P2_GeometryFrequency - Chapter P2: Equilibrium Geometry and...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chapter P2: Equilibrium Geometry and Vibrational Frequencies Introduction Two classes of theoretical models, Hartree-Fock models and density functional models, have now been defined. We now use these models to calculate two closely-related molecular properties, equilibrium geometry and vibrational frequencies. More than anything else, geometry defines a molecule, whereas vibrational frequencies not only form the basis of infrared spectroscopy but, as we shall see in Chapter P3, furnish correction to calculated energies to allow them to be related to measured heats (enthalpies) and Gibbs energies. The first objective of this chapter will be to assess the behavior of HartreeFock and B3LYP density functional models with very large “limiting” basis sets with regard to both equilibrium geometries and vibrational frequencies. This will allow us to “remove” (or at least minimize) the effects of the LCAO approximation and concentrate on problems caused by the Hartree-Fock approximation and the success of density functional models in eliminating these problems. The second objective will be to examine the same two classes of models but with smaller, more practical basis sets. After we establish the smallest basis set that reliably reproduces limiting behavior, we will be able to extend assessment to larger molecules and to use the calculations to explore chemistry. 1 Equilibrium Geometry The three-dimensional structure or geometry of a molecule and with its energy are arguably its most important characteristics. An equilibrium geometry is needed in order to calculate all other molecular properties (including the energy), and providing geometry is normally the first step in any theoretical investigation. Before we examine methods for determining and verifying geometry, we review the sources and quality of experimental geometries. This will provide standards of reliably for experimental data and therefore serve as a guide to judging how well the different classes of theoretical models actually perform. Bond distances and angles of many if not most molecules (at least organic molecules) may easily be anticipated, at least qualitatively. For example, carbon-carbon single bonds generally fall in the range of 1.45–1.55Å, and within this range vary predictably with the types of hybrids involved. This leads to the possibility of establishing the 3D geometry of a molecule based only on its Lewis structure. Given the computational cost involved in obtaining an equilibrium geometry, it is quite legitimate to ask if this is effort well spent. Of course, there are situations where a Lewis description may be problematic, for example, where “correct” Lewis structure is not apparent or where two or more Lewis structures may contribute. A much more important concern, however, is that Lewis structures are too “coarse grain” to allow subtle differences among molecules to be revealed. 2 Sources and Quality of Experimental Equilibrium Geometries There is a wealth of information about equilibrium geometry, accumulated over nearly a century by a variety of experimental techniques. We shall focus on only two of these techniques: microwave spectroscopy which is carried out on a gaseous sample and is arguably the most precise of the available methods, and X-ray diffraction which is carried out on a solid crystalline sample and is by far the most widely used of the available methods. Other experimental techniques, in particular, infrared spectroscopy and electron diffraction in the gas phase and neutron diffraction in the solid, are also able to provide information about molecular structure. They have certainly proven to be of value in some instances, but because they have been less commonly employed they will not be addressed. Microwave Spectroscopy Equilibrium geometries for over a thousand small molecules in the gas phase have been determined by microwave spectroscopy [xxx]. The experiment involves excitation of rotational energy states by radio frequency radiation and leads directly to the three principal moments of inertia for each unique set of atomic masses. Perhaps the most interesting present-day application of microwave spectroscopy is radio astromony. Much of what we know about the existence of molecules in interstellar space comes from assigning the holes in “white radiation” in the radiofrequency range to molecules in space. The incredibly low concentration of matter in interstellar space (on the order of x molecules/m3) is offset by an incredibly long path length. A catalog of “holes” (microwave absorptions) exists and has been used to positively identify several hundred molecules. Many “holes” have yet to be assigned, meaning that there are more molecules to be discovered. A more familiar application of “microwave spectroscopy” is the microwave oven. This uses a source of radiation specifically tuned to one of the rotational lines of the water molecule. Absorption causes water molecules (present in most foods) to heat up. Unfortunately, microwave spectroscopy is now only rarely used for structure determination. In part, this is because it has been supplanted by theoretical calculations, which for molecules of the size amenable to experimental investigation leads to geometries of comparable quality, at far lower expense. Also many if not most of the “easy” molecules for microwave spectroscopy have already been investigated. We need to emphasize, however, that microwave data remains the best source of high-quality experimental data for the structures of polyatomic molecules. In the absence of symmetry, the equilibrium geometry for a molecule with N atoms involves a total of 3N-6 bond lengths and angles, and its complete determination requires as many as N-2 individual spectral measurements with 3 different isotopic substitutions. A few elements occur naturally with sizable populations of more than one isotope, for example, chlorine with two isotopes in a ratio of 76:24 and bromine with two isotopes in a ratio of 51:49. Here, a microwave spectrum will yield multiple sets of moments of inertia. However, most elements, notably hydrogen, carbon, nitrogen and oxygen exist predominately as a single isotope. Here, the synthesis of specifically labeled compounds is necessary. At best, this is tedious. At worst, it is synthetically difficult and very expensive. It is not surprising, therefore, that the use of the technique has for the most part been limited to very simple (and often highly symmetric) molecules. In summary, the primary advantage of microwave spectroscopy is its precision. Bond lengths are often accurate to within ±0.005Å. Many view microwave spectroscopy as the “gold standard” of experimental methods for obtaining molecular geometries. The primary disadvantage of microwave spectroscopy is the need to examine multiple isotopic substitutions, that is, the need to prepare multiple molecules with different isotopes. A second serious disadvantage is the requirement of a permanent dipole moment. Many very small molecules, for example, all homonuclear diatomic molecules, do not exhibit a microwave spectrum. Molecules with finite but very small dipole moments, for example, saturated hydrocarbons, will show only weak microwave transitions which may be difficult to observe. Should we get into the difference between re, r0, etc? X-Ray Diffraction Equilibrium geometries for more than 600,000 small to medium size organic, inorganic and organometallic molecules [xxx] as crystalline solids have been established by X-ray diffraction. In addition, the structures or partial structures of more than 40,000 proteins and protein-small molecule aggregates [xxx] and several thousand minerals and “materials” [xxx] have been established. Crystallography has moved into the mainstream for determining the geometries of “new molecules” (in particular, new organic molecules), and has become the main source of information about the shapes of proteins and protein-small molecule complexes. Except for proteins and other macromolecules, bond lengths are typically accurate to ±0.02-0.04Å. (Individual bond distances in proteins can be in error by 1Å or more.) X-ray diffraction generally provides a poor account of the positions of hydrogen atoms. The primary reason is, as previously mentioned in Chapter P1, that there are relatively few electrons associated with hydrogen atoms. A related problem is that bonds to heavy atoms are often too short by as much as 0.24 0.3Å, due to the fact that the electron density around hydrogen is shifted toward the atom to which it is bonded. This is illustrated below for acetylene. Note, however that this is a systematic error, meaning that it may easily be taken into account. The fact that very small molecules typically do not easily crystallize (at least at “normal” temperatures), and molecules that do crystallize may be too large for full structure determinations with microwave spectroscopy, means that the three-dimensional geometries of only a relatively few molecules are known both in the gas and in the solid. This makes it difficult to assess the effect of the crystalline environment on geometry. Some differences for some molecules are to be expected in order to benefit most from intermolecular interactions, for example, intermolecular hydrogen bonds, and more generally from the demands of packing in the crystal. However, the limited data that are available suggest that changes in bond lengths and angles in going from the gas to the crystal are likely to be modest, and even inside the error bounds of the experimental method. Changes in conformation (torsional or dihedral angles) from the gas to the crystal are likely to be larger and more common. This will be addressed in Chapter P5. The primary advantage of X-ray diffraction it is easily applied to a wide variety of molecules and (unlike microwave spectroscopy) requires a single sample. The primary disadvantage is that this sample must be a crystalline solid. Some molecules will not crystallize (or will not produce “good” crystals) and their geometries cannot be determined by X-ray diffraction. This includes many small (low molecular weight) molecules which may be ideal candidates for microwave spectroscopy. Thus, the two experimental techniques are in one sense complementary. X-ray diffraction lacks the precision of microwave spectroscopy. Bond lengths in medium sized molecules are seldom established to better than ±0.02Å (and are commonly known to no better than twice this amount). Hydrogen positions are poorly identified and bond lengths to hydrogen are commonly too short by 0.1 Å or more. Finally, intermolecular interactions associated with the need to pack into a crystal, may influence bond lengths and angles. 5 Obtaining and Verifying an Equilibrium Structure An equilibrium structure or equilibrium geometry is a minimum in all dimensions on a multi-dimensional potential energy surface. There will likely be many such locations on such a surface. In chemical terms, these energy minima may correspond to different isomers or to different conformers of a particular isomer. For example, cis and gauche-1-butene and cyclobutane are three of the many minima on the C4H6 potential energy surface. The first two are conformers (related by twisting about a single bond) and (collectively) are isomers of cyclobutane (related by bond reorganization). CH3 H C C H H H H H C C C H H C H C H3 H HH C C HH HH C C HH In order to qualify as an equilibrium structure, two mathematical requirements must be met. First, the structure must correspond to a stationary point, meaning that all first derivatives on the 3N-6 dimensional (N atoms) potential surface (the gradient of the energy) must be zero. ∂E(x)/∂xi = 0 for all xi Second, all terms in the diagonal representation of the matrix of second derivatives (the Hessian) must be positive. In order to verify this, the full matrix of second energy derivatives (∂2E(x)/∂xi∂xj) in the original coordinates first needs to be assembled and then replaced by a new set of coordinates (socalled normal coordinates, ζ) such that the matrix second energy derivatives in these coordinates is diagonal. ∂2E( ζ) /∂ ζ i∂ ζ j = δij[∂2E( ζ) /∂ ζ i2] δij is the Kronecker delta function (1 if i=j; 0 otherwise). It will turn out that the set of second energy derivatives in normal coordinates relate to the infrared and Raman frequencies (this will be addressed later in this chapter). For the moment, let us say only that positive (diagonal) Hessian elements correspond to minima on the energy surface and lead to vibrational frequencies that are real numbers, while negative elements correspond to maxima on the surface and lead to vibrational frequencies that are imaginary numbers. If a vibrational frequency is imaginary, the coordinate motion 6 corresponding to this frequency can be animated pointing the way down to an energy minimum. Satisfaction of these two mathematical requirements does guarantee that a particular structure corresponds to an equilibrium geometry, but does not guarantee that this structure can actually be experimentally detected, let alone isolated and characterized. This requires in addition that there are no easily accessible pathways leading to species that are significantly more stable. Discussion will be provided in Chapter P4. In practice, obtaining an equilibrium structure involves an iterative process starting with a ”guess” at the geometry and terminating only when all first derivatives fall below a preset tolerance, assuming that neither the energy nor the atomic coordinates have changed significantly from their values in the previous iteration. The required number of iterations will typically be of the same order as the number of independent variables, although this will depend on how close the guess geometry is to the final geometry. The reason for this is that each iteration in an optimization involves calculation of an energy gradient. Assuming a quadratic energy surface, calculation of 3N-6 gradients (one/iteration) is sufficient to project to the energy minimum. While such a procedure is costly in terms of overall computation (an order of magnitude or more than an energy calculation), it can be fully automated. Determining an equilibrium geometry is no more difficult in terms of “human effort” than calculating a property for a fixed geometry. Because of computational cost, calculation of the full Hessian (the “difficult” part) and evaluation of vibrational frequencies from this Hessian (the “easy” part) is generally not undertaken. Rather, the approximate second derivative matrix formed during course of seeking a zero gradient point (see box above) is employed by the optimization procedure. If desired, the Hessian and vibrational frequencies can be calculated following geometry optimization. Most programs that automatically determine geometry make use of molecular symmetry. The original motive was to save computer time. This is no longer as relevant not only because computers are now much faster, but more so because only small molecules generally possess elements of symmetry. Note, however, that once established (or set) symmetry elements will be maintained. This means that calculated geometries of molecules with symmetry elements may not necessarily be energy minima. A good example is 7 provided by ammonia. Starting from a planar (D3h symmetry) structure will not give rise to the pyramidal (C3v symmetry) equilibrium geometry. Rather, a planar structure will result. . Equilibrium Structures Have Real Frequencies: Obtain equilibrium geometries for the chair and boat forms of cyclohexane using the HF/6-31G* model and calculate vibrational frequencies. Are all frequencies real for both molecules? Which if either molecule is not a minima on the energy surface? Elaborate. Equilibrium Geometry of Disilylene (H2Si=SiH2): Use the B3LYP/6-31G* model to calculate the equilibrium geometry of disilylene, the simplest molecule incorporating a silicon-silicon double bond. Assume a planar (ethane-like) structure. Calculate vibrational frequencies. Are all real numbers? If not, perform the following operations: First, animate any imaginary frequencies to see how disilylene wants to distort to move it to a energy minimum. Next, distort your structure accordingly. Finally, reoptimize the geometry (of the distorted molecule) and again calculate vibrational frequencies. Repeat your calculations and analysis for digermene, H2Ge=GeH2, the simplest molecule with a germanium-germanium double bond. Are the two systems similar with regard to their planarity? CSD results 8 “Limiting” Behavior of Hartree-Fock and Density Functional Models for Equilibrium Geometries We first set out to establish (or at least to estimate) the limiting behavior of Hartree-Fock and density functional models with regard to equilibrium geometries. This will allow us to separate the effects of the LCAO approximation from effects arising from replacement of the exact manyelectron wavefunction by an approximate Hartree-Fock or density functional wavefunction. While it is not possible or at least not practical to actually reach the limit, it is possible to use a sufficiently large basis set such that the addition of further functions to the basis should have only a small effect on calculated equilibrium geometry. The cc-pVQZ basis set will be employed. This is about as large a basis set that be applied for geometry calculations on molecules with more than a few non-hydrogen (“heavy”) atoms. We refer to lithium through neon as first-row elements and sodium through argon as second-row elements. Hydrogen and helium may be thought of as comprising the “zeroeth” row. For a first-row element, the cc-pVQZ basis set comprises a core made up of 8 s-type Gaussians, and a valence split into four parts made up of 8,1,1,1 s–type Gaussians and 3,1,1,1 sets of p-type Gaussians. Three sets of d-type Gaussians, two sets of f-type Gaussians and a set of g-type Gaussians are then added. The related smaller cc-pVTZ basis set (a core made up of 7 s-type Gaussians, and a valence split into three parts made up of 7,1,1 s–type Gaussians and 3,1,1 sets of p-type Gaussians, supplemented by two sets of dtype Gaussians and a set of f-type Gaussians) is employed to establish the extent to which geometries change in response to additional basis functions. It is also not practical to explore all the different functionals that have been proposed. We limit ourselves to a single functional, specifically the B3LYP functional. This is probably the most commonly employed and thoroughly documented functional. Finally, it is not practical to explore the behavior of “limiting” Hartree-Fock and B3LYP models with the cc-pVQZ basis set for all types of molecules. As elaborated in Chapter X, analytical forms of some of the integrals arising in density functional methods are not available and numerical integration procedures are required. This requires that a “grid” of integration points be defined, introducing another set of variables (overall number of grid points and the location of each point) into the calculation. The B3LYP density calculations employed here make use of the so called SG1 grid [xxx]. Experience suggests that equilibrium geometries (and other properties) obtained using this 9 grid are very close to those obtained using much larger (and computationally much more costly) grids. Errors less than 0.005Ǻ and 0.1o are expected. We restrict ourselves to relatively small molecules comprising first and second-row main group elements only. Full details have been provided only for one-heavy-atom hydrides and for hydrocarbons, although summaries in the form of Excel spreadsheets have been provided for a variety of other types of molecules. Establishing equilibrium geometries for molecules with transition metals and for most intermolecular complexes is not practical with a basis set as large as cc-pVQZ, primarily because of the size of the molecules involved. Geometries obtained from Hartree-Fock and B3LYP models with smaller basis sets will be compared with experimental structures later in this chapter. Signed deviations in A-H bond lengths (in Ǻ x 1000) calculated using the Hartree-Fock cc-pVQZ and B3LYP cc-pVQZ models (HF/cc-pVQZ and B3LYP/cc-pVQZ, respectively) for one-heavy-atom hydrides appear alongside of experimental distances in Table P2-1. Unsigned deviations from calculations using the smaller cc-pVTZ basis set in lieu of cc-pVQZ are also provided. This allows us to judge to what extent the cc-pVQZ basis set actually represents a limit. An Excel spreadsheet containing AH bond distances for both Hartree-Fock and B3LYP models with both cc-pVTZ and cc-pVQZ basis sets is provided on the CD-ROM accompanying this text ( AH bond distances). Except for the hydrides of lithium and sodium, “limiting” Hartree-Fock bond lengths are consistently shorter than experimental values. This trend and the two exceptions to it can easily be rationalized by recalling the discussion of the full configuration method as a means to correlate the motions of electrons and improve on the Hartree-Fock model (see Chapter X). Full configuration interaction involves a weighted mixing of all possible excited states with the ground state. One way to think about this is in terms of promotion of one or more electrons from occupied molecular orbitals in the ground-state (HartreeFock) wavefunction to unoccupied molecular orbitals. The lower the promotion energy, the more likely it is that the resulting excited state will contribute to the overall mix. Therefore, the most important contributions should be those in which an electron is removed from one of the few highestoccupied molecular orbitals and placed in one of the few lowest-unoccupied molecular orbitals. 10 Table P2-1: Signed Errors in “Limiting” Hartree-Fock and B3LYP A-H Bond Lengths in Hydrogen and in One-Heavy-Atom Hydrides AHn (Å x 1000) Hartree-Fock molecule to expt to cc-pVTZ B3LYP to expt to cc-pVTZ expt. H2 -8 0 0 0 0.742 LiH CH4 NH3 H2O HF 10 -10 -14 -18 -20 1 0 1 1 1 -7 -5 1 2 4 1 1 1 1 1 1.596 1.092 1.012 0.958 0.917 NaH SiH4 PH3 H2S HCl 33 -6 -15 -9 -9 0 3 3 3 1 -10 -2 -2 7 8 7 3 2 2 1 1.887 1.481 1.420 1.336 1.275 11 The two highest-occupied molecular orbitals of hydrogen fluoride are the same energy (they are said to be degenerate) and correspond to 2p atomic orbitals on fluorine. They are not involved in bonding to hydrogen and electron removal from either or both should have little if any consequence on the hydrogen-fluorine bond distance. Underneath these two orbitals is a σ bonding orbital, electron removal from which should lead to bond lengthening. The lowest-energy unoccupied molecular orbital is antibonding between hydrogen and fluorine. Electron promotion into this orbital, as would occur in the full CI treatment, should lead to weakening (lengthening) of the hydrogen-fluorine bond. Taken together, this implies the unambiguous conclusion that the hydrogen-fluorine bond distance in the limiting HartreeFock model (prior to full CI) is too short. This is in fact exactly what is observed. The situation is different (and ambiguous) for lithium hydride. Here both the highest-occupied and lowest-unoccupied molecular orbitals are LiH bonding. Electron removal from the former and should result in bond weakening (lengthening), while addition of electrons to the latter should result in bond strengthening (shortening). Thus, it is not clear whether the effects of a full CI treatment will be to lengthen or shorten the bond, and it is not clear whether a limiting Hartree-Fock calculation will yield too short or too long a bond. In fact, the calculated bond lengths for both LiH and NaH are longer than experimental values. “Limiting” B3LYP calculations lead to bond distances that are much closer to experimental values. The mean absolute error is 0.004Ǻ compared to 0.014Ǻ for the corresponding Hartree-Fock calculations. Comparison with experimental data does not reveal a systematic error in B3LYP bond lengths. However, with the exception of lithium hydride and sodium hydride, B3LYP/cc-pVQZ bond distances are longer than HF/cc-pVQZ lengths. 12 Bond lengths from both Hartree-Fock and B3LYP calculations with the smaller cc-pVTZ basis set are very similar to those obtained using the larger cc-pVQZ basis set. Differences are 0.001Ǻ or less for first-row hydrides but larger for hydrides involving second-row elements. This suggests that while the cc-pVQZ basis set is sufficiently large to account for limiting behavior of hydrides with first row elements, it may lack the flexibility needed to deal with molecules containing heavier elements. Carbon-Fluorine and Carbon-Lithium Bond Lengths: Is the trend in calculated bond distances for HF and LiH bonds repeated for CF and CLi bonds? The experimental CF bond distance in methyl fluoride is 1.383Ǻ, but the distance in methyl lithium is not known (the molecule exists as a tetramer). To decide, examine equilibrium geometries for methyl fluoride and methyl lithium that have previously been obtained from both Hartree-Fock and B3LYP models with the cc-pVQZ basis set. These are found in methyl fluoride and methyl lithium on the Spartan Student CD. Bond Angles in Ammonia and Water: Examine the bond angles in ammonia and water that have previously been obtained from both Hartree-Fock and B3LYP models with the cc-pVQZ basis set. These are found in ammonia and water on the Spartan Student CD. How do they compare with the experimental HNH bond angle (xxx o) and HOH bond angle (xxx o)? Are you able to rationalize the differences between calculated Hartree-Fock and B3LYP bond angles using the kinds of orbital arguments previously applied for bond lengths? Elaborate. A particularly interesting comparison involving carbon-carbon bond distances in hydrocarbons is provided in Table P2-2. “Limiting” Hartree-Fock lengths for double and triple bonds are shorter than experimental values by an average of 0.024Ǻ, with individual deviations ranging from 0.017Ǻ to 0.028Ǻ. An Excel spreadsheet containing hydrocarbon bond distances for both Hartree-Fock and B3LYP models with both cc-pVTZ and cc-pVQZ basis sets is provided on the CD-ROM accompanying this text (hydrocarbon bond distances). Orbital shapes again provide rationale for the trends. For example, the fact that the highest-occupied molecular orbital of ethylene is strongly CC bonding while the lowest-unoccupied molecular orbital is strongly CC antibonding, suggests that electron promotion from the former to the latter would lead to bond lengthening, that is, the Hartree-Fock bond length) must be too short. 13 Table P2-2: Signed Errors in “Limiting” Hartree-Fock and B3LYP Bond Lengths in Hydrocarbons (Å x 1000) Hartree-Fock to B3LYP to to cc-pVTZ expt to molecule expt triple bonds acetylene propyne but-1-yne-3-ene -24 -25 -26 1 1 1 -8 -8 -6 1 1 0 1.203 1.206 1.208 double bonds butatriene C2C3 allene butatriene C1C2 methylenecyclopropane propene ethylene but-1-yne-3-ene cyclobutene 1,3-butadiene -24 -17 -19 -28 -21 -25 -24 -23 -24 0 1 0 1 0 0 1 1 0 -20 -8 -7 -17 -10 -15 -7 -8 -9 0 0 1 0 0 0 0 1 0 1.283 1.308 1.318 1.332 1.336 1.339 1.341 1.342 1.343 aromatic bonds benzene -15 1 -7 1 1.397 single bonds but-1-yne-3-ene methylenecyclopropane C2C3 propyne 1,3-butadiene bicyclo[1.1.0]butane C1C3 bicyclo[1.1.0]butane C1C2 propene cyclopropane cyclobutene C2C3 propane ethane methylenecyclopropane C3C4 cyclobutane cyclobutene C3C4 5 1 5 -3 -11 -34 -2 -15 -4 -1 -7 -15 -5 -5 0 1 0 0 1 0 0 1 1 0 0 0 1 0 -11 -5 -4 -14 11 -3 -4 -5 -2 1 -4 -5 3 3 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1.431 1.457 1.459 1.467 1.497 1.498 1.501 1.510 1.517 1.526 1.531 1.542 1.548 1.566 14 cc-pVTZ expt. Single bond distances are better described from the “limiting” Hartree-Fock model than are the lengths of double and triple bonds, although the errors span a larger range. Most bond lengths are sorter than experimental values (by as much as 0.034Ǻ for the bridging bond in bicyclo[1.1.0]butane), although some bonds are actually longer. Arguments based on the relative bonding/antibonding character of the highestoccupied and lowest-unoccupied molecular orbitals are not always successful in anticipating whether Hartree-Fock single bond lengths are likely to be too short or too long. 1,3-butadiene provides a particularly simple example. The HOMO is clearly antibonding between the two center carbons while the LUMO is clearly bonding. It might be expected, therefore, that transfer of electrons should lead to bond strengthening (shortening), meaning the limiting Hartree-Fock single bond distance should be should be too long. In fact, the “limiting” Hartree-Fock bond length is 0.003Ǻ longer than the corresponding B3LYP distance and only slightly (0.003Ǻ) shorter than the experimental value. The B3LYP/cc-pVQZ model also provides a uniform account of double and triple bond lengths in hydrocarbons. All calculated lengths are smaller than experimental values and the mean absolute error for the set is 0.010Ǻ (less than half that for the corresponding Hartree-Fock model). Bond lengths from both Hartree-Fock and B3LYP models with the cc-pVTZ basis set are nearly identical from those from the corresponding cc-pVQZ models. The differences here are smaller than those previously noted for bond distances in one-heavy-atom hydrides. This suggests that the convergence of both Hartree-Fock and B3LYP models depends to some extent on the kind of bond. 15 Similar results and similar interpretations follow from comparisons of other classes of compounds. A summary of mean absolute deviations of bond lengths obtained from Hartree-Fock and B3LYP models with the cc-pVQZ basis set for diatomic molecules, molecules with bonds between carbon and a heteroatom, molecules with bonds between two heteroatoms and hypervalent molecules (in addition to one-heavy-atom hydrides and hydrocarbons) is provided in Table P2-3. Excel spreadsheets containing bond distances for both Hartree-Fock and B3LYP models with both cc-pVTZ and cc-pVQZ basis sets for diatomic molecules, molecules with carbon-heteroatom bonds, molecules with heteroatom-heteroatom bonds and in hypervalent molecules are provided on the CD-ROM accompanying this text (bond distances in diatomic molecules, carbon-heteroatom bond distances, heteroatom-heteroatom bond distances, and bond distances in hypervalent molecules, respectively). Carbon-Carbon Bond Lengths in Cyclopropane and Cyclobutane: The experimental carbon-carbon bond length in cyclopropane is 0.016Ǻ shorter than that in propane whereas the bond distance in cyclobutane is 0.022Ǻ longer. Provide a rationale for this. 16 Table P2-3: Mean Absolute Deviations of HF/cc-pVQZ and B3LYP/ccpVQZ Bond Lengths Obtained from Experimental Distances and Bond Lengths Obtained from HF/cc-pVTZ and B3LYP/cc-pVTZ Models (Å) Hartree-Fock class of compounds B3LYP from expt. from cc-pVTZ one-heavy-atom hydrides 21 1 6 2 carbon-carbon carbon-heteroatom heteroatom-heteroatom diatomic molecules hypervalent molecules 15 20 37 34 25 1 2 3 3 5 9 8 16 10 16 0 2 4 4 6 17 from expt. from cc-pVTZ Practical Hartree-Fock and B3LYP Models for Equilibrium Geometry Calculations Except for very small molecules, Hartree-Fock and B3LYP calculations using the cc-pVTZ and cc-pVQZ basis sets will not be practical for determining equilibrium geometries. Large basis set calculations are of value to judge the limits and ultimately the quality of the underlying models, but smaller basis sets are needed for routine applications to larger molecules, at least at present. Here, we ask if equilibrium geometries obtained from Hartree-Fock and B3LYP calculations using the smaller 6-31G* and 6-311+G** basis sets are able to reproduce experimental geometries to an “acceptable” level of accuracy. Our criterion for bond lengths is ±0.02Å, somewhat less than the best experimental data (0.005Ǻ for microwave spectroscopy) and comparable if not somewhat better than that from commonly-available data (0.02-0.04Ǻ from X-ray diffraction). For a first-row element, the 6-311+G** basis set comprises a core made up of 6 s-type Gaussians, and a valence split into three parts made up of 3,1,1 s–type Gaussians and 3,1,1 sets of p-type Gaussians, s and p Gaussians. A single a set of d-type Gaussians and set of diffuse Gaussians is added. Hydrogens are represented by a two s-type Gaussians and a set of p-type Gaussians. The smaller 6-31G* basis set also comprises a core made up of 6 stype Gaussians, but the valence is split into only two parts made up of 3 and 1 s–type Gaussians and 3 and 1 sets of p-type Gaussians, supplemented by a set of d-type Gaussians. Hydrogens are represented by two s-type Gaussians. Signed deviations from experimental A-H bond lengths (Ǻ x 1000) for oneheavy-atom hydrides from Hartree-Fock and B3LYP methods with 6-31G* and 6-311+G** basis sets are provided in Table P2-4. An Excel spreadsheet containing AH bond distances for both Hartree-Fock and B3LYP models with 6-31G*, 6-311+G**, cc-pVTZ and cc-pVQZ basis sets is provided on the CDROM accompanying this text (AH bond distances). In terms of mean absolute error, all four models actually meet the criterion, and only for LiH and NaH are bond lengths in error by more than 0.02Ǻ. Bond lengths from HF/6-311+G** calculations are better than those from HF/6-31G* calculations and those from B3LYP/6-311+G** calculations are better than those from B3LYP/6-31G* calculations. As expected, HartreeFock models yield bond lengths that are consistently shorter than experimental values. On the other hand, bond lengths from B3LYP calculations are typically larger than experimental values. An unexpected (and probably fortuitous) result is that the B3LYP bond lengths appear to be (slightly) less sensitive to basis set than Hartree-Fock lengths. 18 Table P2-4: Signed Deviations of Hartree-Fock and B3LYP A-H Bond Lengths in Hydrogen and in One-Heavy-Atom Hydrides from Experimental Bond Lengths (Å x 1000 ) molecule Hartree-Fock 6-31G* 6-311+G** B3LYP 6-31G* 6-311+G** expt. H2 -12 -7 1 2 0.742 LiH CH4 NH3 H2O HF 42 -8 -9 -11 -6 11 -8 -12 -17 -20 25 1 7 11 17 -4 -1 3 4 5 1.596 1.092 1.012 0.958 0.917 NaH SiH4 PH3 H2S HCl -30 -6 -17 -9 -9 -31 -4 -13 -5 -6 -3 5 4 13 15 -3 3 3 12 13 1.887 1.481 1.420 1.336 1.275 mean absolute error 14 12 9 5 – 19 Bond lengths in hydrocarbons offer more subtle criteria with which to judge the different practical models. While triple bonds are not sensitive to structure, the carbon-carbon double bonds provided in Table P2-5 below show a range of 0.06Ǻ (from the central bond in butatriene to the bond in 1,3-butadiene) and single bonds a range of more than 0.13Ǻ (from the bond in but-1-yne-3ene to the C3C4 bond in cyclobutene). In terms of mean absolute error, all four models are satisfactory with both B3LYP models being superior to both Hartree-Fock models. With a few notable exceptions, trends in carbon-carbon bond distances are well reproduced. Similar results and similar interpretations follow from comparisons of other classes of compounds. A summary of mean absolute deviations of bond lengths obtained from the two Hartree-Fock models and the two B3LYP models for diatomic molecules, molecules with bonds between carbon and a heteroatom, molecules with bonds between two heteroatoms and hypervalent molecules (in addition to one-heavy-atom hydrides and hydrocarbons) is provided in Table P2-6. Excel spreadsheets containing bond distances Excel spreadsheets containing bond distances for both Hartree-Fock and B3LYP models with 6-31G*, 6-311+G**, cc-pVTZ and cc-pVQZ basis sets for diatomic molecules, molecules with carbon-heteroatom bonds, molecules with heteroatom-heteroatom bonds and in hypervalent molecules are provided on the CD-ROM accompanying this text (bond distances in diatomic molecules, carbonheteroatom bond distances, heteroatom-heteroatom bond distances, and bond distances in hypervalent molecules, respectively). 20 Table P2-5: Deviations from Experiment of Hartree-Fock and B3LYP Bond Lengths in Hydrocarbons (Å) Hartree-Fock molecule 6-31G* 6-311+G** B3LYP 6-31G* 6-311+G** expt. triple bonds acetylene propyne but-1-yne-3-ene -18 -19 -19 -20 -21 -21 2 1 3 -4 -4 -2 1.203 1.206 1.208 double bonds butatriene C2C3 allene butatriene C1C2 methylenecyclopropane propene ethylene but-1-yne-3-ene cyclobutene 1,3-butadiene -18 -12 -15 -24 -18 -22 -19 -20 -22 -21 -13 -16 -23 -16 -21 -19 -18 -21 -12 -1 0 -10 -3 -8 0 -1 -4 -17 -5 -3 -12 -5 -10 -3 -3 -6 1.283 1.308 1.318 1.332 1.336 1.339 1.341 1.342 1.345 aromatic bonds benzene -11 -11 0 -2 1.397 single bonds but-1-yne-3-ene methylenecyclopropane C2C3 propyne 1,3-butadiene bicyclo[1.1.0]butane C1C3 bicyclo[1.1.0]butane C1C2 propene cyclopropane cyclobutene C2C3 propane ethane methylenecyclopropane C3C4 cyclobutane cyclobutene C3C4 mean absolute error 8 5 9 1 -30 -10 2 -12 -2 2 -4 -15 -3 -3 13 7 6 7 0 -25 -6 2 -10 0 2 -4 -12 -2 -3 12 -7 13 1 -10 -4 2 1 -1 2 6 0 -3 6 6 4 -9 10 1.431 1.457 1.459 1.467 1.497 1.498 1.501 1.510 1.517 1.526 1.531 1.542 1.548 1.566 – 21 -11 -4 3 -1 -1 2 6 0 0 6 7 5 Table P2-6: Mean Absolute Deviations from Experiment of Hartree-Fock and B3LYP Bond Lengths (Å) Hartree-Fock class of compounds one-heavy-atom hydrides carbon-carbon carbon-heteroatom diatomic molecules hypervalent molecules 6-31G* B3LYP 6-311+G** 14 13 12 12 22 6-31G* 6-311+G** 9 4 5 5 Can Single Bonds be Shorter than Double Bonds? Suggest a hydrocarbon that incorporates a single bond as short as possible, and another (or the same) hydrocarbon that incorporates a double bond as long as possible. Justify your selection. Obtain the equilibrium geometry for your molecule using the B3LYP/6-311+G** model. Are the bond lengths outside the range of those presented in Table P2-2? If they are not, refine your choice of molecules. Have you managed to uncover a single bond that is shorter than a double bond? Water Dimer: The water dimer exhibits a structure with a single hydrogen bond and an OO separation of 2.98Ǻ. H O H O H H Both Hartree-Fock and B3LYP models with the cc-pVQZ basis set show similar overall geometries, but with different OO distances of 3.03Ǻ and 2.91Ǻ, respectively. Do these accurately represent the limits of the Hartree-Fock and B3LYP methods? To tell, obtain equilibrium geometries for water dimer using the HF/6-311+G** and B3LYP/6-311+G** models. 23 Using Calculated Equilibrium Geometries Combination of Hartree-Fock and B3LYP Hamiltonians with the 6-31G* and 6-311+G** basis sets leads to four of the simplest available models for the calculation of equilibrium geometries for small to medium size molecules (up to 50 heavy atoms). They complement the available experimental techniques, in the sense of being able to handle molecules larger than possible (or at least practical) with microwave spectroscopy, and produce results that are more accurate (or at least as accurate) as those from X-ray crystallography. Of course, many interesting molecules cannot easily be synthesized (or synthesized at all), and here calculations may be the only choice. At the extreme, a molecule needs to exist only in the mind of the chemist. Hartree-Fock and B3LYP density functional models are unbiased in that they have not been parameterized to reproduce experimental data, and it is not unreasonable to expect that they will perform as well for molecules for which experimental data are unavailable as they will for molecules for which data exist. Cations, anions, radicals, hydrogen-bonded complexes among other short-lived species may be investigated as easily as “normal” molecules. By the same token, lack of experimental data complicates assessment of the models. The structures of ions in particular may not be obtained microwave spectroscopy, and experimental data will generally be limited to solid phase structures. Here, different counterions or different ion-counterion arrangements may lead to different ion geometries. It could be argued that the some among the present generation of density functional models are biased in the sense that they have been extensively parameterized to match experimental data. Discussion is provided in Chapter X. Whether they are used in conjunction with experimental methods or completely on their own, these four theoretical models offer chemists a convenient way to probe the geometries of molecules, real or imagined. In doing so, they provide a means to assess the validity of existing qualitative models for chemical bonding, for example, the VSEPR model, as well as to formulate and test new models. Because a molecule need not exist in order for its structure to be examined, the range of calculations exceeds that of experimental chemistry. 24 Microwave Spectra of Ions: Why can’t microwave transitions of charged molecules be observed? CH Bonds and Hybridization at Carbon: Obtain equilibrium geometries for ethane, ethylene and acetylene using the HF/6-31G* model. Do CH bond lengths change with the hybridization at carbon? If they do, is the magnitude of the changes (in terms of a percentage) similar to that for CC bonds (see Table P2-5)? Diborane: Is it better to depict diborane with or without a boron-boron bond? H B H B H H H B B H H H To decide, examine a density surface for diborane corresponding to 50% enclosure of the total number of electrons. Is the surface in the middle of the “bond” convex (suggesting buildup of electron density) or concave (suggesting depletion of density)? For comparison, examine the corresponding surface for ethylene. Use the HF/6-31G* model to first obtain equilibrium geometries for the two molecules. Diborane was originally thought to look like ethane. Obtain the geometry of ethane-like borane (D3d symmetry) and calculate vibrational frequencies. Is this structure an energy minimum? If not, provide a rationalization. Hint: Examine the highest-occupied orbital(s) of ethane, and ask what would happen were two electrons to be removed. Structure of Sulfur Tetrafluoride: VSEPR (Valence State Electron Pair Repulsion) theory uses two simple rules is able to assign geometry. The first is that the geometry about an atom is determined by insisting that electron pairs (either lone pairs or bonds) avoid each other as much as possible. An atom surrounded by two electron pairs assumes a linear geometry, by three pairs a trigonal-planar geometry, four pairs a tetrahedral geometry, five pairs a trigonal-bipyramidal geometry and six pairs an octahedral geometry. The second rule is that it is more important to avoid unfavorable lone pair-lone pair interactions than it is to avoid lone pair-bond interactions which are in turn more important to avoid than bondbond interactions. According to the first rule, the the sulfur in sulfur tetrafluoride with five electron pairs (four bonds and a lone pair) assumes a trigonal bipyramidal geometry. According to the second rule, the lone pair will prefer to occupy an equatorial rather than axial position. This means that SF4 adopts a see-saw geometry in which the lone pair is 90° to two of the SF bonds and 120° to the other two bonds, rather than a trigonal pyramidal geometry in which all three bonds are 90° to the lone pair. F F F F F F S• • S •• F F "see saw" trigonal pyramid 25 Use the HF/6-31G* model, to obtain geometries for both see-saw (C2v symmetry). and trigonal pyramid (C3v symmetry) forms of SF4. Is the see-saw structure lower in energy than the trigonal-pyramid structure in accord with VSEPR theory? Are both structures energy minima? Elaborate. If they are, is the energy difference between them small enough that both would be seen at room temperature? CaF2. A Failure of VSEPR Theory? According to VSEPR theory, CaF2 should be a linear molecule. However, in the solid phase the molecule is bent. Is this a failure of the VSEPR model or is it merely a consequence of crystal packing? Staring with a bent structure, obtain the equilibrium geometry of CaF2 using the B3LYP/6-31G* model. Is the molecule bent? If it is not, then calculate an equilibrium geometry for a molecule that is constrained to having a FCaF bond angle of 140o and compare its energy to that of "linear" CaF2. What does the energy difference tell you about the magnitude of the crystal packing energy? If on the other hand, "free" CaF2 is bent, then calculate the equilibrium geometry of "linear" CaF2 and compare its energy to that of the bent molecule. Geometry Change s with Change in the Number of Electrons: A molecule's geometry depends not only on the constituent atoms, but also on the total number of electrons. Use the HF/6-31G* model to obtain equilibrium geometry for 2-methyl-2- propyl cation (tertbutyl cation), as well as those for the corresponding radical (with one additional electron) and the anion (with two additional electrons). Describe any changes to the geometry of the central carbon with increasing number of valence electrons, and speculate on the origin of these changes. Radical Cation of Diborane: Diborane incorporates what can only be described as a π bonding orbital analogous to the familiar π bond in ethylene. Removal of an electron from this orbital should result in elongation of the boron-boron bond, just as removal of an electron from the π orbital in ethylene results in elongation of the CC bond. However, unlike ethylene, the π orbital in diborane is not the HOMO but rather an orbital of lower energy. This suggests that the radical cation of diborane (formed from ionization of diborane) might be quite different than the radical cation of ethylene. Use the B3LYP/6-31G* model to obtain the equilibrium geometry of diborane and display the HOMO. Predict what should happen to the geometry of diborane were an electron to be 26 removed from this orbital. Test your prediction by calculating the equilibrium geometry of radical cation or diborane. Make certain that you start with a distorted (C1 summetry) geometry. Compare BB bond lengths for diborane and its radical cation. Protonated Alkenes and Alkanes: Protonation of a molecule with a localized electron pair leads to a new σ bond. Its geometry is easy to anticipate by analogy with neutral molecules containing the same number of electrons. For example, protonated ammonia and protonated trimethylamine are expected to incorporate a tetrahedral (nitrogen) center, just as their neutral isoelectronic counterparts, methane and isobutane incorporate tetrahedral (carbon) centers. The geometry of a protonated alkene is less clear. For example, is the proton in protonated ethylene primarily associated with a single carbon does it “bridge” both carbons? H H H C + C H H H+ HC H C H H Calculate equilibrium geometries for both open and bridged forms of protonated ethylene (ethyl cation) using the B3LYP/6-31G* model. Do both structures appear to be minima on the C2H5+ potential surface or does one of the structures “collapse” to the other? Elaborate. If there is only one energy minimum, is it open or bridged? Even alkanes protonate in the gas phase, even though they incorporate no obvious electronrich sites. Use the B3LYP/6-31G* model to explore possible structures for protonated methane (CH5+). Calculate vibrational frequencies for whatever you uncover to verify it is actually and energy minimum. Describe the bonding in terms of a weak complex or a molecule with a pentavalent carbon. Repeat you calculations and analysis for protonated ethane (C2H7+). 2-Norbornyl Cation: 2-Norbornyl cation ranks among the most studied and controversial molecules in 20th century chemistry. Literally hundreds of papers found their way into the organic chemical literature, and prompted a lively and sometimes vitriolic debate between two future Nobel laureates on what became known as the “non-classical ion problem”. The observation that led to the debate was that C2 and C6 positions in norbornane substituted in the 2 position by a good (anionic) leaving group scrambled. This could be accommodated either by invoking a rapid equilibration between two “classical” cations in which the “positively charged” carbon is tricoordinate, or insisting that there was only a single “non-classical” cation incorporating a pentacoordinate carbon. 27 The issue was finally settled by a series of beautiful experiments by George Olah, the most important being the low-temperature proton and 13C NMR spectra of the ion. The latter is shown below. What is the structure of 2-norbornyl cation? Use the B3LYP/6-31G* model to calculate its geometry, infrared and 13C NMR spectra. Start from a “classical” structure. How do you know that what you have found is an energy minimum? Does the calculated 13C spectrum fit what is observed? If so, are the experimental spectral assignments in line with the calculations? Elaborate. Dicyclopentadienyl Beryllium: Ferrocene exhibits a beautiful structure in which iron is sandwiched between two cyclopentadienyl rings. The usual way of writing this is to give the iron a formal +2 charge and each cyclopentadienyl ring unit negative charge. Why would such an arrangement be expected to be especially stable? -1 Fe+2 -1 It may come as a surprise, then that dicyclopentadienyl beryllium does not look like ferrocene at all. Rather, it adopts a half-sandwhich structure, with one cyclopentadienyl ring above the metal (as in ferrocene) but with the other σ bonded to beryllium. Be Be 28 Use the B3LYP/6-31G* model to obtain geometries and vibrational frequencies for the sandwich and half-sandwich structures of dicyclopentadienyl beryllium. Is the half sandwich preferred? Are both energy minima? Elaborate. Provide an explanation for the change in geometry. Carbon-Fluorine Bond Lengths in Fluorosilanes and Fluorogermanes: As discussed earlier in this chapter, carbon-fluorine bond lengths in fluoromethanes, CFnH4-n (n=1-4), decrease dramatically with increasing number of fluorines, from 1.xxǺ in fluoromethane to 1.xxǺ in tetrafluoromethane. Is the same trend found in the corresponding fluorosilanes, SiFnH4-n (n=1-4)? Perform B3LYP/6-31G* calculations to tell. If so, is the percentage bond length change smaller, larger or of comparable magnitude? Repeat your calculations and analysis for the fluorogermanes, GeFnH4-n (n=1-4). Bond Angles in Amines and Ethers: Bond angles about nitrogen in amines and about oxygen in ethers are typically close to tetrahedral, For example, measured bond angles in trimethylamine and dimethylether are xxxo and yyyo, repectively. Replacement of methyl by something bulkier, for example, a tert-butyl group, might be expected to lead to an increase in non-bonded (steric) repulsion and result in an increase in bond angle. Use the B3LYP/6-31G* model to obtain equilibrium geometries for trimethylamine and tri-(tertbutyl)amine using the B3LYP/6-31G* model. Do you observe the expected increase in CNC bond angle? Are there any other conspicuous structural changes between the two molecules, in particular, have the CN bonds lengthened? Repeat your calculations for dimethyl ether and di-(tert-butyl) ether. Is there an increase in COC bond angle? Replacement of methyl by a group capable of drawing electrons away from the lone pair on nitrogen (two lone pairs on oxygen), for example, a silyl group, might also be expected to result in an increase in bond angle. Why? Obtain the equilibrium geometry for trisilylamine. Is the SiNSi bond angle larger than the CNC bond angle in trimethylamine? Is it as large as the CNC bond angle in tri-(tert-butyl)amine? Repeat your calculations and answer analogous questions for disilyl ether. 29 Lewis Structures and Equilibrium Geometries At the start of this chapter, we implied that a Lewis structure together with a table of “standard” bond lengths and angles is not likely to provide a good enough account of molecular geometry to allow subtle differences in energies or other properties to be uncovered. Even so, Lewis structures are an important part of a chemists’ vocabulary, not only because the offer a very concise way to depict molecular structure but also they provide clues about “interesting” structures. Benzene provides a good example of the latter. Here, there are two equivalent ways of placing three single and three double bonds in a six-membered ring. The fact that we cannot decide which placement is “correct”, implies that neither is “correct”. Rather, the proper description of the geometry of benzene follows from using both. This in turn suggests that the six carbon-carbon bonds in benzene are all identical and midway in length between normal single and double bonds. This is of course exactly what is observed. Lewis Structure for D iazomethane: Diazomethane is usually described as a composite of two Lewis structures, both of which involve separated charges. + N – – N + N N Obtain the geometry of diazomethane using the HF/6-31G* model. Also obtain the geometries of methylamine, CH3 NH2, and methyleneimine, H2C=NH, as examples of molecules incorporating normal CN single and double bonds, respectively, and of trans diimide, HN=NH, and nitrogen, N≡N, as examples of molecules incorporating normal NN double and triple bonds, respectively. Which Lewis structure provides the better description for diazomethane or are both required for adequate representation? Examine electrostatic charges for diazomethane. Do they suggest the same Lewis structure as the bond distances? S tructure of Ozone: Suggest two different Lewis structures for ozone, O3. (One or both may require non-zero formal charges.) Obtain the equilibrium geometries corresponding to 30 both s tructures using the B3LYP/6-31G* model. Which structure is lower in energy? Is it in accord with the experimentally known equilibrium geometry? Calculate the infrared s pectrum of the higher-energy structure to establish whether or not it is an energy minimum? Explain you reasoning. If the preferred structure has more than one distinct oxygen atom, which is most positively charged? Most negatively charged? Is your result based on electrostatic charges consistent with that based on formal charges? Which Lewis Structure? Anthracene and Phenanthrene: Whereas the two Lewis structures for benzene are equivalent and thus need to be weighted equally, the situation is less clear where the Lewis structures are different. For example, two of the three Lewis structures that can be written for naphthalene are the same but the third is different. In this case, any conclusions regarding molecular geometry depend on the relative weight given to each structure. Assigning equal weights to all three structures suggests that four of the bonds in naphthalene (that are double bonds in two of the three Lewis structures) should be shorter than the remaining seven bonds, (that are double bonds in only one of the three Lewis structures). This is in fact what is observed experimentally. 1.42Å 1.43Å 1.42Å 1.38Å Draw the complete set of Lewis structures for anthracene and phenanthrene. anthracene phenanthrene Assuming that each Lewis structure contributes equally, assign which if any of the carboncarbon bonds should be especially short and which if any should be especially long. Next, obtain equilibrium geometries for the two molecules using the HF/6-31G* model. Are your assignments consistent with the results of the calculations? If not, suggest which Lewis structures need to be weighed more heavily (or which need to be weighed less heavily) in order to bring the two sets of data into accord. 31 Molecules with Transition Metals Equilibrium geometries for ~200,000 compounds incorporating transition metals are known experimentally, almost entirely from X-ray crystallography. While this is similar to the number of structures for main-group compounds that are available, relatively little attention has been given to the calculation of geometries for transition-metal inorganic and organometallic compounds. In part, this is no doubt due to the well-known failure of Hartree-Fock models to provide satisfactory geometries for molecules with transition metals. However, as shown Table P2-7 for a small selection of organometallic carbonyl compounds, the B3LYP/6-31G* model provides a reasonable successful account. In particular, metal-carbon bond lengths are typically reproduced to within 0.02Ǻ (the experimental error). Transition metal inorganic and organometallic chemistry represents an attractive target for quantum chemistry. 32 Table P2-7: Comparison of Metal-Carbon Bond Distances in Organoiron Compounds from B3LYP/6-31G* and Experiment (Å) organoiron compounds bond B3LYP/6-31G* ferrocene expt. 2.05 2.06 butadiene iron tricarbonyl butadiene C1 butadiene C2 CO 2.10 2.07 1.78 2.14 2.06 1.76 cyclobutadiene iron tricarbonyl cyclobutadiene CO 2.04 1.78 2.06 1.79 ethylene iron tetracarbonyl ethylene COaxial COequatorial 2.12 1.81 1.79 2.12 1.78 1.81 acetylene iron tetracarbonyl acetylene COaxial COequatorial 2.08 1.82 1.79 2.08 1.77 1.76 33 Carbon Monoxide as a Ligand: Carbon monoxide is perhaps the most common ligand in transition-metal organometallic compounds. CO molecule acts is to donate an electron pair into an empty orbital on the metal atom. In return, electrons are donated from an occupied orbital on the metal atom into low-lying empty orbitals on the CO molecule. Obtain the equilibrium geometry for carbon monoxide using the B3LYP/6-31G* model. Display the HOMO. Is it bonding, antibonding or essentially non-bonding between carbon and oxygen? What, if anything, would you expect to happen to the CO bond strength as electrons are donated from the HOMO to the metal atom? Display the LUMO. (There are actually two equivalent LUMOs, designated LUMO and LUMO+1, and you can base arguments on either one.) Is the LUMO bonding, antibonding or essentially non-bonding between carbon and oxygen? What if anything would you expect to happen to the CO bond strength if electrons were donated from the metal atom into this orbital? Elaborate. Will this affect the change that results from electrons being donated from the HOMO of the CO molecule to the metal atom? Elaborate. Examine the molecular orbitals of Fe(CO)4, arising from loss of CO from Fe(CO)5 to see if the metal center possesses a high energy filled molecular orbital of proper symmetry to donate electrons into the LUMO of carbon monoxide. Obtain the equilibrium geometry for Fe(CO)5 using the B3LYP/6-31G* model, then delete one of the equatorial CO ligands to make a Fe(CO)4 fragment. Calculate the energy of the fragment (don’t optimize the geometry) . Display the HOMO. Does the HOMO in Fe(CO)4 have significant amplitude in the location where the carbon monoxide ligand will attach? If so, does it have the proper symmetry to interact with the LUMO in CO? Would you expect electron donation to occur? Elaborate. Binding Ethylene to a Metal: Two limiting structures can be drawn to represent ethylene bonded to a transition metal. The first may be thought of as a weak complex in that it maintains the carbon-carbon double bond, while the second destroys the double bond in order to form two new metal-carbon σ bonds, leading to a three-membered ring (a metallacycle). Most likely, real metal-alkene complexes will exhibit bonding intermediate between these two extremes. M M Optimize the geometry of ethylene using the B3LYP/6-31G* model and examine both the HOMO and LUMO. Is the HOMO bonding, antibonding or non-bonding between the two carbons? What if anything should happen to the carbon-carbon bond as electrons are donated from the HOMO to the metal? Do you expect the carbon-carbon bond length to decrease, increase or remain about the same? Elaborate. The LUMO is where the next (pair of) electrons will go. Is this orbital bonding, antibonding or non-bonding between the two carbons? What, if anything, should happen to the carbon-carbon bond as electrons are donated (from the metal) into the LUMO? Is the expected change in the carbon-carbon bond due to this interaction in the same direction or in the opposite direction as any change due to interaction of the HOMO with the metal? Elaborate. 34 Optimize the geometry of ethylene iron tetracarbonyl using the B3LYP/6-31G* model. Compare the carbon-carbon bond to that in ethylene? Based on geometry, how would you describe the bonding between the ethylene and the metal. CO CO Fe CO CO Next, delete the ethylene ligand and calculate the energy calculation on the resulting (iron tetracarbonyl) fragment (don’t optimize the geometry). Examine both the HOMO and LUMO of this fragment. Is the LUMO of the iron tetracarbonyl fragment properly situated to interact with the HOMO of ethylene? Elaborate. Would you expect electron donation from ethylene to the metal to occur? Does the HOMO of the fragment have the proper symmetry to interact with the LUMO of ethylene? Elaborate. Would you expect electron donation from the metal to ethylene to occur? Is Chromium Tricarbonyl a π Donor or π Acceptor? Chromium tricarbonyl complexes to one of the faces of benzene leaving the other face exposed for further reaction. Cr OC CO CO Is there a significant change in the geometry of benzene as a result of complexing to chromium tricarbonyl? In particular, is there evidence of bond localization? To decide, use the B3LYP/6-31G* model to calculate equilibrium geometries for both benzene and benzene chromium tricarbonyl. Does the Cr(CO)3 group act to donate electrons leading to enhanced affinity toward electrophiles or to accept electrons leading to diminished reactivity? Compare electrostatic potential maps for “free” and complexed benzene. To establish a point of reference, include electrostatic potential maps (based on equilibrium geometries from the B3LYP/6-31G* model for (“free”) aniline (an electron-rich arene) and nitrobenzene (an electron-poor arene). You need to make certain that all four maps are on the same scale. Do you see the expected trend in electrostatic potential in the maps for benzene, aniline and nitrobenzene? Elaborate. Where does benzene chromium tricarbonyl fit in? Classify the Cr(CO)3 group as an electron donor or acceptor. Rank the chromium tricarbonyl group relative to either the amino group or the nitro group as appropriate. Benzene Chromium Tricarbonyl vs. Borazine Chromium Tricarbonyl: Borazine, B3N3H6, is often considered to be a close analogue to benzene in that it contains six π electrons in a planar six-membered ring. However, all the electrons (formally) come from the three nitrogens, suggesting instead that it may not be as “delocalized” as benzene. Like benzene, borazine forms complexes with chromium tricarbonyl. B N B B OC Cr CO CO N N Cr CO OC CO 35 Use the B3LYP/6-31G* model to calculate the geometry of borazine and borazine chromium tricarbonyl. Does borazine undergo a significant change in geometry as a result of binding to the metal carbonyl? How does this compare to the change in the geometry of benzene resulting from its complexation to chromium tricarbonyl (see previous problem)? Is there evidence of bond localization? If it does, is this bond polarized toward titanium or toward carbon? Titanium-Carbon Double Bonds: While double bonds to heavy main-group carbon analogues are quite rare, double bonds between carbon and the transition metals, titanium, zirconium and hafnium are common. Use the B3LYP/6-31G* model to obtain the equilibrium geometry of bis-cyclopentadienyl titanium methylidene, along with that of tetramethyltitanium as a reference. Me Ti Me Cp Me Ti Me CH2 Cp Is the shrinkage in TiC bond length from single to double comparable to that typically found for hydrocarbons, either in absolute terms or on a percentage basis? Elaborate. Display the HOMO of bis-cyclopentadienyl titanium methylidene. Does it incorporate a π bond? bis-Cyclopentadienyl Titanium Ethylidene: Use the B3LYP/6-31G* model to obtain the equilibrium geometry of bis-cyclopentadienyl titanium ethylidene. H Cp Ti C Me Cp Is the titanium-carbon bond length shorter, longer or about the same as the bond in biscyclopentadienyl titanium methylidene? If it is significantly different, provide a rationale as to why. (Hint: examine the TiCMe and TiCH bond angles.) Transition Metal Carbyne Complexes: While molecules that incorporate a metal-carbon double bond (“carbenes”) are commonplace, molecules with a metal-carbon triple bond (“carbynes”) are less frequently encountered. Use the B3LYP/6-31G* model to obtain the equilibrium geometry for the chromium carbyne shown below. For comparison, also obtain the geometry for propyne. OC Br OC CO Co C CH3 CO Can you identify three metal-carbon binding molecular orbitals? If yes, are these orbitals qualitatively similar to those in propyne? Elaborate. Is the carbon-carbon single bond in the metal carbyne short, that is, does it reflect sp hybridization at carbon? Compare it to the analogous CH bond length in propyne. 36 Hybridization and Bond Lengths to Transition Metals: Single bonds to sp2 hybridized carbon centers are shorter than those to sp3 centers but not as short as those to sp centers. The differences can be attributed to the fact that valence 2p orbitals extend further from their atomic centers than the analogous 2s orbital. The same principle should extend to (transition) metal-ligand bonding. Here, the hybrids comprise primarily (n) d and (n+1) stype orbitals (The corresponding (n+1) p orbitals are usually assumed to play a lesser role.) Use the B3LYP/6-31G* model to obtain equilibrium geometries for the titanium methyl (X2YTi-CH3, methylidene, XYTi=CH2, and methylidyne, XTi≡CH complexes, where X=Y=H. X Y X Ti CH3 Y T i CH 2 Y Ti CH Y Compare Ti-H bond lengths for the three compounds. Is there significant variation? If there is, rationalize this behavior on the basis of the nature (hybridization) of the orbitals on the metal. The three compounds you have examined are extremely electron deficient, far short of the 16-18 electrons normally needed to satisfy titanium. Any effects that this might have on their structures can be tempered by substituting one of the hydrogen “ligands” by a Cp (cyclopentadienyl) ligand. Carry out calculations on the methyl and methylidene systemw with X=Cp, Y=H. Is the same trend present in these compounds? If it is, is the magnitude of the change about the same or is it increased or diminished. Rationalize your result. Repeat both sets of calculations replacing H with a methyl and then with chlorine. Do the trends that you observed for H maintain? 37 Hydrogen-Bonded and Related Complexes In the limit of weak metal-ligand bonding, transition-metal inorganic and organometallic compounds may be viewed as intermolecular complexes. That is to say that the components, the metal fragment and one or more ligands, maintain their essential identity. There are many other classes of molecules of this type, most familiar and arguably most important among them being hydrogen-bonded complexes. Here, one molecule incorporates an atom with a non-bonded electron pair. It is referred to as the hydrogen-bond acceptor and it is the electron donor. The other molecule, referred as the hydrogen-bond donor, incorporates a bond between hydrogen and an electronegative atom. Because of the difference in electronegativities, the σ molecular orbital describing this bond is primarily localized on the electronegative atom, while the corresponding σ* molecular orbital is primarily localized on hydrogen. Therefore, the hydrogen is electron deficient and should act as an electron acceptor. Interaction of the filled molecular orbital on the hydrogen-bond acceptor (electron donor) and the empty molecular orbital on the hydrogenbond donor (electron acceptor) is the driving force behind the formation of a stable complex. Most commonly, hydrogen-bond acceptors are nitrogen or oxygen atoms and hydrogen-bond donors involve bonds to nitrogen or oxygen. Halogens and sulfur may also act as (weak) hydrogen-bond acceptors and SH and PH bond as (weak) hydrogen-bond donors. Consider water dimer. The common terminology refers to the molecule on the left as a hydrogen-bond acceptor and the molecule on the right as a hydrogenbond donor. Alternatively, one might describe the molecule on the left as an electron-pair donor and the molecule on the right as an electron-pair acceptor. Note that the water molecule is able to act as both a hydrogen-bond acceptor and hydrogen-bond donor. In fact, the same molecule may serve simultaneously in both roles in hydrogen-bonded complexes. Liquid water offers a good example, with each water molecule contributing up to two electron pairs (hydrogen-bond acceptors) and two OH bonds (hydrogen-bond donors). This leads to a three dimensional network of hydrogen bonds and gives liquid water its unique properties. A small water cluster is depicted below. 38 Table P2-8 compares hydrogen bond lengths for several complexes obtained from HF/6-31G* and B3LYP/6-31G* models with experimental values. An Excel spreadsheet containing hydrogen-bond distances for these and other complexes from both Hartree-Fock and B3LYP models with 6-31G*, 6-311+G* and cc-pVTZ basis sets is provided on the CD-ROM accompanying this text (bond distances in hydrogenbonded complexes). 39 Table P2-8: Comparison of Hartree-Fock and B3LYP Connecting Molecules in Intermolecular Complexes with Experimental Values (Å) complex geometrical parameter HF/6-31G* B3LYP/6-31G* expt. water dimer OO 2.97 2.98 hydrogen fluoride dimer FF 2.71 2.79 acetic acid dimer OO 2.79 2.66 2.68 adenine-thymine NO 3.08 2.94 2.97 NN 3.01 2.88 2.89 ON 2.92 2.81 2.94 NN 3.06 2.96 2.93 NO 3.00 2.92 2.80 guanine-cytosine mean absolute error 40 Alternative Structure of the Water Dimer: Water molecule incorporates two electron pairs that may act as hydrogen bond acceptors and two OH bonds that may act as hydrogen-bond donors. This means that it should be possible to construct a dimer with two hydrogen bonds. H H O HO H Attempt to obtain an equilibrium geometry for the doubly hydrogen-bonded structure of water dimer. Use the B3LYP/6-31G* model. If you find such a structure, confirm that it is or is not an energy minimum. If it is an energy minimum, calculate the room-temperature Boltzmann distribution of the two different forms of water dimer. Hydronuim cation: Use the B3LYP/6-31G* model to calculate the equilibrium geometry of hydronium cation, H3O+. Guess a structure for the complex between H3O+ and both four water molecules, and obtain its geometry with the B3LYP/6-31G* model. Point out any significant changes that have occurred to the cation a result. In particular, is there evidence for “sharing” the proton? Is the positive charge primarily localized on the hydronium cation or has it spread out to the surrounding water molecules? Chloride Anion: Use the B3LYP/6-31G* model to calculate the equilibrium geometry for chloride anion surrounded by four water molecules. Is the negative charge primarily localized on chlorine or has it spread out to the water molecules? Hydrogen Positions in Acetic Acid Dimer: Acetic acids forms a symmetrical hydrogenbonded dimer. HO O H3C C C O H CH3 O Calculate equilibrium structures for acetic acid and its dimer using the HF/6-31G* model. Point out any significant changes in bond lengths and angles. Have the hydrogens involved in the hydrogen bonds moved to positions halfway between the oxygens or have they remained with one oxygen (as in acetic acid)? Do the structural changes (or lack of structural changes) suggest that hydrogen bonds are comparable to normal (covalent) bonds or are they weaker? Elaborate. Repeat your calculations for trifluoroacetic acid and its dimer. Does the fact that it is a much stronger acid that acetic acid translate into larger structural changes upon dimeization? Elaborate. Borane Carbonyl: Borane carbonyl, BH3CO, may be viewed to result from interaction of the non-bonded electron pair (on carbon) in carbon monoxide and an empty p-type molecular orbital (on boron) in borane. H H H B C O 41 Borane carbonyl is different from transition-metal carbonyls in that there is little possibility for significant back bonding, that is, interaction of a high-lying filled molecular orbital on borane with an empty π* molecular orbital on carbon monoxide. This suggests that the CO bond in carbon monoxide should not change significantly as a result of complexion. Use the B3LYP/6-31G* model to obtain equilibrium geometries for both carbon monoxide and borane carbonyl. Is there a significant change in CO bond length? If there is, how does it compare with bond length changes previously for the equatorial and axial CO groups in iron pentacarbonyl (see earlier problem in the chapter)? Repeat your calculations and analysis for trifluoroborane carbonyl, BF3CO. Identify any significant difference between the structure of this and borane carbonyl. Intramolecular Gallium-Amine Complexes: Main-group elements below boron in the periodic table generally perfer planar trigonal structures, for example, trimethylaluminum and trimethyl gallium. Me Me Al Me Me Me Ga Me However, molecules such as these (as well as analogous boron compounds) are known to form donor-acceptor complexes with amines. Here, the nitrogen acts as the electron donor and the aluminum or gallium as the acceptor. Me Me Me Al Me Me Me Me Me N Ga Me Me Me N Me Use the HF/6-31G* model to obtain equilibrium geometries for the two complexes shown above. To what extent does electron transfer actually occur from the donor to the acceptor? Are either or both of the complexes better represented a zwitterions? Elaborate. A good example of a molecule where the desire to be planar and the benefit of complex formation come into conflict is the tricyclic gallium-amine complex shown below. Is this an “open” structure allowing the CCC bond angles to be nearly tetrahedral, or a “closed” structure allowing donor-acceptor interaction, or both? Ga N Use the Hartree-Fock 6-31G* model to decide. First, obtain an equilibrium geometry starting from an open structure, and then (if necessary) start from a closed structure. Do you find one or two structures? If two structures, which is more stable? If one structure, is it open or closed? 42 Vibrational Frequencies In addition to the equilibrium geometry, another important quantity that may be directly obtained from examination of a potential energy surface is the set of vibrational frequencies. We have already used frequencies to verify that a particular point on an energy surface in fact corresponds to an energy minimum, and we will use them later to connect calculated energy to measure enthapies and Gibbs energies (Chapter P3) and to verify transition-state geometries (Chapter P4). Here, we establish the connection between the vibrational frequencies of a molecule and its infrared spectrum. One-Dimensional Systems In one dimension, calculation of the vibrational frequency for a (diatomic) molecule first requires that the energy be expanded in terms of a Taylor series. E(x) = E(x0) + (dE(x)/dx) x + (d2E(x)/dx2) x2 + higher-order terms The harmonic approximation assumes that only the second derivative term needs to be considered. This term is commonly known as the force constant, and indicates the relative ease or difficulty of moving away from the equilibrium structure, that is, the curvature of the surface at the minimum. E(x0) in the Taylor expansion is a constant, dE(x)/dx (the gradient) is zero at the equilibrium geometry. (Note that the gradient will not be zero away from the minimum, making this expression invalid.) Cubic and higher-order terms (so-called anharmonic terms) are ignored. The square root of the ratio of the second-derivative term and the reduced mass (the product of the masses of the two atoms divided by their sum) is proportional to the frequency. vibrational frequency √ [(d2E(x)/dx2/)reduced mass] A small second derivative means that distortion away from the equilibrium position is “easy” and leads to a low frequency, while a large second derivative means that distortion is “difficult” and leads to a high frequency. Signed errors (in cm-1) for measured vibrational frequencies for diatomic molecules obtained from “limiting” (cc-pVQZ basis set) Hartree-Fock and 43 B3LYP models are provided in Table P2-9. Also tabulated are (unsigned) deviations between calculations with the cc-pVTZ and cc-pVQZ basis sets. Except for molecules with lithium and sodium, all Hartree-Fock vibrational frequencies are significantly larger than experimental values. This parallels the trend in calculated bond lengths and follows from our previous discussion of what might be expected from excitation of electrons from filled to unfilled molecular orbitals. Frequencies obtained from the B3LYP model are typically much closer to experimental frequencies, although most are still larger. Differences in vibrational frequencies from models with the cc-vTQZ and smaller cc-pVTZ basis sets are generally much smaller than differences with experimental frequencies for both Hartree-Fock and B3LYP models. Many-Dimensional Systems Generalization to polyatomic molecules is straightforward. The energy is expanded in the same way as before, the only difference being that a vector quantity, x, replaces a scalar quantity, x. E(x) = E(x0) + Σi(∂E(x)/∂xi)xi + ½Σij(∂2E(x)/∂xi∂xj)xixj + higher-order terms The dimension of x is 3N for a molecule with N atoms. However, the number of vibrational frequencies is 3N-6 (3N-5 for a linear molecule), the remaining 6 (5) dimensions corresponding to translation away from and rotation around the center of mass. Calculation of the vibrational frequencies involves three steps. In the first step, the set of 3N x 3N second energy derivatives with respect to the Cartesian coordinates is obtained. The energy second derivatives then need to be mass weighted. Diagonal terms (∂2E(x)/∂xi2) are divided by the mass of the atom associated with xi, and off-diagonal terms (∂2E(x)/∂xi∂xj) are divided by the product of the square root of the masses of the atoms associated with xi and xj. These expressions reduce to that already provided for the onedimensional case. In the second step, the original (Cartesian) coordinates are replaced by a new set of coordinates (so-called normal coordinates) such that the matrix of massweighted second energy derivatives is diagonal. [∂2E( ζ) /∂ ζ i∂ ζ j]/(√Mi√Mj) = δij [∂2E( ζ) /∂ ζ i2]/Mi 44 Table P2-9: Signed Errors (Experiment – Calculated) in “Limiting” (ccpVQZ Basis Set) Hartree-Fock Vibrational Frequencies for Diatomic Molecules and Unsigned Deviations Between Frequencies Obtained from cc-pVTZ and cc-pVQZ Basis Sets (cm-1) Hartree-Fock molecule B3LYP to expt to cc-pVTZ to expt to cc-pVTZ expt. LiF -48 6 -8 3 898 LiCl 0 3 -3 3 641 CO -284 2 -73 2 2143 N2 -399 2 -117 4 2331 O2 -391 18 -53 18 1580 F2 -372 3 -155 7 891 FCl -133 2 -9 2 784 NaF -3 9 32 21 536 NaCl 8 1 5 2 363 Cl2 -54 1 30 0 560 45 δij is the Kronecker delta function (1 if i=j; 0 otherwise). In the third step, the six coordinates corresponding to the three translations and three rotations are removed. This leaves 3N-6 internal vibrational motions. Anharmonic Effects Because the actual potential energy in the vicinity of the minimum has been approximated by a quadratic function, calculated frequencies will almost always be larger than measured values. This is reasonable because a quadratic function goes to infinity with increase in distance rather than going asymptotically to a constant (separated atoms). At least in principle, it is possible to estimate the effect of anharmonic contributions to measured vibrational frequencies, that is, to extract harmonic frequencies. This may be accomplished by comparing the actual spacing of the energy levels associated with the ground and excited states of a particular vibration (which go to zero with increasing level) with the constant spacing associated with a quadratic (harmonic oscillator) potential. In practice, such an analysis can only be done for diatomic and for very simple polyatomic molecules. An Excel spreadsheet containing vibrational frequencies for hydrogen and one-heavy-atom hydrides calculated from “limiting” (cc-pVQZ basis set) Hartree-Fock and B3LYP models, both with directly measured frequencies and with “harmonic frequencies” extracted from 46 the experimental spectra is provided on the CD-ROM accompanying this text (vibrational frequencies of one-heavy-atom hydrides). All frequencies are larger than measured values, and lead to a mean absolute errors of 225 cm-1 (~6%) and 140 cm-1 (~4%) for Hartree-Fock and B3LYP models, respectively. Most frequencies are also larger than the experimental harmonic values. Mean absolute errors are significantly reduced; 140 cm-1 (~4%) for the Hartree-Fock model and 38 cm-1 (~1%) for the B3LYP model. It is also possible to account for anharmonic effects by removing the restriction that cubic and higher-order terms are assumed to be zero. In practice perturbation theory is used to approximate an expansion of the energy through fourth order. E(x) = E(x0) + Σi[∂E(x)/∂xixi] + ½Σij[∂2E(x)/∂xi∂xj]xixj 3 4 1/6Σijk[∂ E(x)/∂xi∂xj∂xkxixjxk ] + 1/24Σijkl[∂ E(x)/∂xi∂xj∂xk∂xl]xixjxkxl + Calculating anharmonic corrections requires one to two orders of magnitude more effort than calculating harmonic frequencies. Because of this, it has been done only for very small molecules. Harmonic Frequencies: Measured frequencies corrected for anharmonicity (harmonic frequencies) are available for a variety of small molecules, including many diatomic molecules. A sampling (in cm-1) include: LiF, 914; CO, 2170; N2, 2360 and F2, 923. For each, calculate the percentage of the total error of the “limiting” Hartree-Fock and B3LYP frequencies given in Table P2-9 due to the harmonic approximation. 47 Practical Hartree-Fock and B3LYP Models for Vibrational Frequency Calculations Except for very small molecules, Hartree-Fock and B3LYP models with large basis sets such as the cc-pVTZ and cc-pVQZ basis sets are not practical for calculation of vibrational frequencies. This is the same situation previously discussed for calculation of equilibrium geometries, but aggravated by the fact that second-energy derivatives are much more difficult and their calculation requires much more computer time than calculation of first derivatives. Therefore, the primary use of cc-pVTZ and cc-pVQZ (and even larger) basis sets is to assess the limits of the underlying (Hartree-Fock and B3LYP) models. Smaller basis sets are needed for practical applications. The performance of two smaller Gaussian basis sets will be examined, 6-311+G** and 6-31G*. Both are easily applicable for molecules with molecular weights up to 300 amu, and the second is applicable for molecules with molecular weights up to 500 amu. The first comparison involves the full set of (measured) frequencies for just four molecules: ethane, methylamine, methanol and methyl fluoride (Table P2-10). The performance of the four “practical” models described above is examined as is that of the Hartree-Fock and B3LYP models with the cc-pVTZ basis set. Errors are quoted in term of a percentage rather than a numerical value. According to this metric, the three Hartree-Fock models are significantly (factor of two-three times) poorer than the corresponding B3LYP models. This is not surprising in view of our previous discussion. Results from both Hartree-Fock and B3LYP models with the 6-311+G** basis set offer significant improvement over the corresponding models with the 6-31G* basis set, and are nearly identical to the “limiting” (cc-pVTZ basis set) values. Together these results suggest that the B3LYP/6-311+G** model is a suitable procedure for (absolute) frequency calculation. A more subtle comparison involves comparisons frequencies associated with similar motions in closely-related molecules. This implies that the vibration of interest is easily identified and that its frequency is well separated from all other frequencies. A good example involves frequencies identified with the stretching of a carbon-carbon double bond in molecules with only a single such bond. Frequencies relative to the CC stretch in ethylene as a standard from Hartree-Fock and B3LYP calculations with the 6-31G*, 6-311+G** and 48 cc-pvTZ basis sets are compared with experimental values in Table P2-11. These span a range (of experimental frequencies) of ~300 cm-1, from the CC stretch in cyclobutene to that in tetrafluoroethylene. The errors noted here are much smaller than those seen previously for absolute frequency comparisons. With only a few exceptions, all models properly order the frequencies for the series of compounds. In terms of mean absolute error, there is little to distinguish the six models 49 Table P2-10: Comparison of Hartree-Fock and B3LYP Vibrational Frequencies for Two-Heavy-Atom Hydrides with Experimental Values (cm-1) symmetry description molecule of vibration of mode CH3CH3 a1g a1u a2u eg eu CH3NH2 a' a'' CH3OH a' a'' CH3F a1 e Hartree-Fock B3LYP 6-31G* 6-311+G** cc-pVTZ 6-31G* 6-311+G** cc-pVTZ expt. CH3 s-stretch CH3 s-deform CC stretch torsion CH3 s-stretch CH3 s-deform CH3 d-stretch CH3 d-deform CH3 rock CH3 d-stretch CH3 d-deform CH3 rock 3203 1580 1060 330 3197 1548 3247 1650 1338 3272 1644 891 3162 1550 1052 331 3154 1517 3201 1618 1320 3227 1615 881 3158 1548 1048 328 3152 1522 3194 1614 1323 3222 1619 881 3047 1456 1011 314 3048 1434 3098 1532 1237 3123 1538 835 3023 1425 996 310 3023 1410 3069 1504 1219 3094 1506 830 3028 1424 996 308 3029 1409 3072 1504 1220 3098 1506 827 2954 1388 995 289 2986 1379 2969 1468 1190 2985 1469 822 NH2 s-stretch CH3 d-stretch CH3 s-stretch NH2 scis. CH3 d-deform CH3 s-deform CH3 rock CN stretch NH2 wag NH2 a-stretch CH3 d-stretch CH3 d-deform NH2 twist CH3 rock torsion 3730 3246 3158 1841 1648 1608 1290 1149 947 3812 3282 1665 1480 1052 342 3737 3201 3122 1801 1618 1582 1265 1138 900 3816 3235 1635 1454 1041 323 3725 3190 3113 1796 1617 1581 1270 1133 912 3803 3221 1635 1458 1041 319 3466 3079 2971 1704 1528 1486 1193 1071 878 3550 3119 1547 1371 988 333 3508 3056 2962 1669 1498 1461 1164 1056 823 3586 3093 1518 1342 974 305 3487 3055 2964 1665 1498 1458 1171 1053 838 3561 3089 1519 1349 976 293 3361 2961 2820 1623 1473 1430 1130 1044 780 3427 2985 1485 1419 1195 268 OH stretch CH3 d-stretch CH3 s-stretch CH3 d-deform CH3 s-deform OH bend CH3 rock CO stretch CH3 d-stretch CH3 d-deform CH3 rock torsion 4117 3306 3186 1663 1638 1508 1189 1164 3232 1652 1290 348 4192 3261 3147 1629 1609 1472 1173 1148 3194 1620 1278 318 4174 3243 3137 1632 1610 1484 1174 1156 3179 1621 1279 318 3754 3133 2997 1541 1511 1401 1097 1067 3040 1525 1183 347 3849 3111 2988 1506 1480 1358 1072 1042 3035 1496 1168 299 3834 3104 2985 1512 1480 1373 1081 1044 3027 1495 1170 309 3681 3000 2844 1477 1455 1345 1060 1033 2960 1477 1165 295 CH3 s-stretch CH3 s-deform CF stretch CH3 d-stretch CH3 d-deform CH3 rock 3233 1652 1187 3314 1653 1312 3193 1611 1156 3277 1615 1296 3180 1619 1173 3256 1619 1306 3037 1532 1091 3111 1524 1205 3032 1480 1033 3117 1490 1185 3025 1489 1058 3102 1494 1194 2930 1464 1049 3006 1467 1182 10 5 mean absolute percentage error 12 9 50 3 3 - Table P2-11: Comparison of Hartree-Fock and B3LYP Relative Carbon-Carbon Double Bond Stretching Frequencies with Experimental Values (cm-1) molecule 6-31G* Hartree-Fock 6-311+G** cc-pVTZ 6-31G* B3LYP 6-311+G** cc-pVTZ expt. cyclobutene -52 -48 -47 -60 -55 -58 -53 tetrachloroethylene -32 -5 -15 -91 -69 -73 -52 -4 0 1 -18 -12 -11 -9 0 0 0 0 0 0 0 cyclopropene 30 34 33 33 43 40 18 propene 25 27 47 19 21 21 33 isobutene 32 35 34 24 27 26 38 tetramethylethylene 55 68 66 31 43 40 60 cis-1,2-difluoroethylene 115 122 120 71 73 74 92 tetrafluoroethylene 289 315 302 208 226 219 249 mean absolute error 13 21 19 21 14 16 – cyclopentene ethylene CC stretching frequencies for ethylene are 1856, 1814, 1820, 1720, 1683 and 1693 cm-1 for Hartree-Fock 6-31G*, 6-311+G** and cc-pVTZ and B3LYP 6-31G*, 6-311+G** and ccpVTZ models, respectively. The experimental stretching frequency is 1623 cm-1. 51 Acetone: Obtain the equilibrium geometry of acetone using the HF/6-31G* model and calculate the infrared spectrum. Locate the line in the spectrum corresponding to the carbonyl stretch. What is the ratio of the Hartree-Fock sketching frequency to the experimental frequency (1731 cm-1)? Locate the line in the spectrum corresponding to the fully symmetric combination of carbon-hydrogen stretching motions. Is the ratio of this frequency to its experimental value (2937 cm-1) similar to that for the ratio of calculated and experimental carbon-oxygen stretching frequencies. Perfluoroacetone: Unlike acetone, the line the infrared spectrum of perfluoroacetone corresponding to the CO stretch is very weak. Obtain equilibrium geometries and infrared s pectra for the two molecules using the B3LYP/6-31G* model. Do the calculations also show a marked decrease in the intensity of the CO stretching frequency from acetone to perfluroracetone? Provide an explanation. Hint: compare the change in dipole moment corresponding to a xx kJ/mol change in energy for motion away from the equilibrium p osition along the normal mode corresponding to the CO stretch. S tretching Frequencies for Bonds Involving Electronegative Atoms: It has previously been pointed out that bond lengths from Hartree-Fock are almost always shorter than experimental values. The magnitude of the error generally increases as the elements involved in the bond move from left to right in the Periodic Table. Thus, the "limiting" CC bond length in ethane is 0.006Ǻ shorter than the experimental value, while the CF bond length in methyl fluoride is 0.025Ǻ shorter and the FF bond length is fluorine is 0.087Ǻ shorter. Is there an analogous trend for stretching frequencies from Hartree-Fock calculations? Using data from Tables P2-9 and P2-10, establish whether there is a correlation between errors in CC, CF and FF stretching frequencies and errors in CC, CF and FF bond distances for ethane, methyl fluoride and fluorine, respectively. Examine both Hartree-Fock and B3LYP models with the 6-311+G** basis set. No calculations are needed. CO Stretching Frequencies in Carbonyl Compounds: CO stretching frequencies span a very narrow range centering around 1750 cm-1. Calculate equilibrium geometries and infrared spectra for trans-acrolein at the low end of the range (CO stretching frequency = 1724 cm-1), methyl formate in the middle of the range (CO stretching frequency = 1754 cm1 ) and acetic acid at the top of the range (CO stretching frequency = 1788 cm-1) using both the HF/6-31G* and B3LYP/6-31G* models. Locate the line in each of the spectra corresponding to the carbonyl stretch. Speculate why this particular infrared absorption is a useful indicator of carbonyl functionality in complex molecules. Do one or both models reproduce the ordering of carbonyl stretching frequencies? If so, which model better accounts for the range in frequency variation? Speculate on what causes the variation. Repeat your calculations using the 6-311+G** basis set instead of 6-31G*. Is there a significant improvement in results? Elaborate. 52 Vibrations of Dimethysulfoxide: Vibrational motions seldom correspond to isolated changes in individual bond lengths or bond angles, but rather involve combinations of these motions. There are of course exceptions, the most notable being the CO stretching motion in carbonyl compounds (see previous problem). Because the vibrational frequency depends on the masses of the atoms involved, any change in frequency resulting from a change the mass (isotope) of one or more atoms can provide insight into the nature of the motion. Calculate the equilibrium geometry and vibrational spectrum of dimethylsulfoxide, (CH3)2S=O, using the HF/6-31G* model. Characterize the motion associated with each infrared frequency as being primarily bond stretching, angle bending or a combination of the two. Is bond stretching or angle bending easier? Do the stretching motions each involve a single bond or do they involve combinations of all three bonds? Next, replace all six hydrogens in dimethylsulfoxide with six deuteriums and recalculate the vibrational spectrum. Compare the resulting frequencies with those obtained for the non-deuterated molecule. Rationalize any differences. CO Stretching Frequencies in Metal Carbonyls: Carbon monoxide is one of the most common ligands in transition metal inorganic and organometallic compounds. Chromium hexacarbonyl, iron pentacarbonyl and nickel tetracarbonyl are representative. Use the B3LYP/6-31G* model to obtain equilibrium geometries for these there carbonyl complexes as well as for “free” carbon monoxide. Is there a significant shift in the frequency of the CO strectch in carbon monoxide as a result of complexation? If there is, offere a rationalization for the direction of the shift. 53 Calculating Infrared Spectra Infrared spectroscopy is one of the most commonly used techniques for identification of molecules. While it does not provide the same level of detail about molecular structure as NMR spectrometry, it requires much simpler, smaller and less expensive instrumentation. This means that it is available to use in environments that are inaccessible to NMR, for example, on the Mars rovers. In addition to the set of vibrational frequencies, calculation of an infrared spectrum requires a matching set of intensities, each of which is proportional to the change in the electric dipole moment for motion along the corresponding vibrational coordinate. formula This means that any vibrational motion that does lead to a change in dipole moment will have a zero infrared intensity and no line in the infrared spectrum will be observed. Extreme cases of this are homonuclear diatomic molecules which do not give rise to an infrared spectrum. More common and more important are vibrational motions that lead to very small changes in dipole moment. Here the line in infrared spectrum may be too weak to be easily observed. Benzyne: Benzyne has long been implicated as a possible intermediate in nucleophilic aromatic substitution, for example, in the reaction of chlorobenzene with strong base. Cl OH OH– OH– –H2 O –Cl– –H2 O benzyne While the geometry of benzyne has yet to be conclusively established, the results of a 13C labeling experiment leave little doubt that two (adjacent) positions on the ring are equivalent. Cl * NH2 * NH2 * KNH2 * + NH3 1 : *= 13 C 1 In addition, the infrared spectrum of benzyne has been recorded and a line in the spectrum at 2085 cm-1 assigned to the C≡C stretch. Obtain equilibrium geometries and calculate infrared spectra for benzyne, 2-butyne and 54 benzene using the B3LYP/6-31G* model. Compare the geometry of benzyne with those of 2-butyne and benzene. Does the molecule incorporate a “real” triple bond (as does 2butyne) or is the length closer to that in benzene? Draw an appropriate Lewis structure (or set of Lewis structures) for benzyne. Locate the line in the infrared spectrum for 2-butyne corresponding to the C≡C stretch. Calculate the ratio of the experimental C≡C stretching frequency in 2-butyne (2240 cm-1) and the calculated value. This will be used to scale the stretching frequency calculated for benzyne. Examine the infrared spectrum for benzyne and decide whether or not it is an energy minimum. Show your reasoning. Locate the line in the spectrum corresponding to the “C≡C” stretch. Is it weak or intense relative to the other lines in the spectrum? Would you expect that this line would be easy or hard to observe? Scale the calculated frequency by the factor you obtained (for 2-butyne) in the previous step. Is your (scaled) stretching frequency in reasonable accord with the reported experimental value of 2085 cm-1? Need to update to current experiments … the old one is incorrect Infrared Spectrum of Acetic Acid Dimer: Acetic acids forms a symmetrical hydrogenbonded dimer. O H3C HO C C O H CH3 O Use the B3LYP/6-31G* model to calculate infrared spectra for acetic acid and its dimer. Point out any significant differences between the frequencies and/or intensities associated with the (two) OH stretching motions in the dimer and the OH stretching frequency in acetic acid. Greenhouse Gases: In order to dissipate the energy that falls on it due to the sun, the earth “radiates” as a so-called “blackbody” into the universe. The “theoretical curve” is a smooth distribution peaking around 900 cm-1 and decaying to nearly zero around 1500 cm-1. This is in the infrared, meaning that some of the radiation will be intercepted by molecules in the earth’s gaseous atmosphere. This in turn means that the earth is actually warmer than it would be were it not to have an atmosphere. This warming is known as the greenhouse effect, to make the analogy between the earth’s atmosphere and the glass of a greenhouse. Both allow energy in and both impede its release. The actual distribution of radiated energy as measured from outside the earth’s atmosphere in the range of 500-1500 cm-1 is given below. The overall profile matches that for a blackbody, but the curve is peppered with holes. 55 Neither nitrogen nor oxygen, which together comprise 99% of the earth’s atmosphere absorbs in the infrared and causes the “holes”. However, several “minor” atmospheric components, carbon dioxide most important among them, absorb in the infrared and contribute directly to greenhouse warming. Its infrared spectrum shows a strong absorption in the region centering 670 cm-1, the location of the most conspicuous hole in the blackbody radiation profile. Identify three of the top 10 chemicals manufactured worldwide. Use the B3LYP/6-31G* model to calculate the infrared spectra for each and comment whether or not you would expect it to be a significant greenhouse gas. 56 Fingerprinting One of the most important if not the most important use of infrared spectra is the identification of compounds through “fingerprinting”. In a sense this takes advantage of the fact that vibrational motions and vibrational energies are not readily transferable from one compound to another. The “fingerprint region” is normally taken to be roughly 600 – 2500 cm-1. Measurement of infrared spectra below this range generally require specialized instrumentation. More importantly, low frequency vibrations often correspond to torsional motions and may depend strongly on detailed conformation. Further discussion is provided in ChapterP5). The region on the spectrum above around 2500 cm-1 typically arises from CH stretching motions and is too crowded to be of much value in distinguishing molecules. discussion of line shape in order to make calculated infrared spectrum recognizable 57 ...
View Full Document

Ask a homework question - tutors are online