Dunning_BasisSets_HF - Solution of the Electronic...

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Unformatted text preview: Solution of the Electronic Schrödinger Equation Using Basis Sets to Solve the HartreeFock Equations Comments on Hartree-Fock Theory Reasonable Description of Electronic Structure Most atoms Many molecules near their equilibrium geometries (although there are exceptions) Basis for More Accurate Calculations HF is zero-order wave function used in many more accurate models of the electronic structure of atoms and molecules that include electron correlation • Configuration interaction • Perturbation theory • Coupled cluster theory Using Basis Sets to Solve the Hartree-Fock Equations Hartree-Fock Wave Function and Energy Just to review, the Hartree-Fock (HF) wave function for Ne electrons is e ˆ (1,2,3,..., Ne ) = A a (1) b (2) c (3) n ( Ne ) a (2) a ( Ne ) b 1 a (1) b (2) b ( Ne ) n = (1) (1) n (2) n ( Ne ) Ne ! where i is a spin orbital, i.e., a combination of a spatial function and a spin function. The energy of the HF wave function is: E= Ne i =1 = Ne i =1 h ii + Ne Ne i =1 j = i +1 h ii + 1 2 Ne Ne i =1 j =1 (J ij K ij ) (J ij K ij ) Using Basis Sets to Solve the Hartree-Fock Equations Hartree-Fock Equations The optimum orbitals to use in the HF wave function are the solutions of the HF equations ˆ f HF i ˆ = [h + Ne k =1 ˆ ( Jk ˆ K k )] i = ii i where ˆ h= 1 2 ˆ J k = dr2 ˆ K k = dr2 2 1 NN One electron operator: (kinetic energy plus nuclear attraction) ZI I =1 rI 1 * (2) k k (2) r12 * ˆ (2) p12 k k r12 (2) Coulomb operator Exchange operator ˆ ( p12 is the permutation operator) Using Basis Sets to Solve the Hartree-Fock Equations RHF Equations for Closed Shell Singlet States For a singlet state with all of the orbitals doubly occupied, Ne = 2n, and the HF wave function is 1 RHF (1,2,3,..., Ne e ˆ )=A aabb nn The HF equations are ˆ f RHF i ˆ = h+ n k =1 ˆ (2 J k ˆ Kk ) i = ii These are the closed shell Restricted Hartree-Fock (RHF) wave function and equations. For atoms and diatomic molecules, the integro-differential RHF equations can be solved by finite difference or finite element methods. For polyatomic molecules, these approaches pose significant mathematical problems and the equations have traditionally been solved using basis set expansion methods. Using Basis Sets to Solve the Hartree-Fock Equations Basis Set Expansion Method The orbital i in the RHF equation is a function of a set of one-electron variables, (x1, y1, z1), and thus it can be represented by an expansion in a basis set: i where = Nbf c =1 i is also a function of (x1, y1, z1). If the basis set, { }, is a complete set, we will obtain an exact solution of the RHF equations. For practical reasons, it is not possible to use a complete basis set, so an error is always associated with the use of this approach. However, as the basis set approaches completeness, this error becomes vanishingly small. Using Basis Sets to Solve the Hartree-Fock Equations Matrix Hartree-Fock Equations Using the basis set expansion method, the RHF equations become ˆ f RHF i ˆ = f RHF Nbf =1 Nbf ˆ f RHF If we now project Nbf =1 dr1 ii = c i =1 i Nbf ci= =1 * μ onto ci= Nbf c i i =1 these equations and integrate, we obtain * μ Nbf =1 ˆ f RHF Nbf ci= R Fμ HF c i = i =1 dr1 * μ c i Nbf i =1 Sμ c i Since these equations must hold for all μ, they reduce to the matrix equation F RHF C = S C Using Basis Sets to Solve the Hartree-Fock Equations Matrix Hartree-Fock Equations (cont’d) In the matrix RHF equations RHF F11 F1RHF K RHF F21 RHF F22 RHF F2 K RHF FK 1 F= RHF F12 RHF FK 2 RHF FKK S11 S1K S21 S22 S2 K SK1 SK 2 S= S12 S KK Fock Matrix Overlap Matrix c11 C= c12 c1K 11 0 0 c21 c22 c2 K 0 22 0 cK 1 cK 2 cKK 0 0 KK Coefficient Matrix = Orbital Energy Matrix (diagonal) Using Basis Sets to Solve the Hartree-Fock Equations Kinetic Energy & Nuclear Attraction Matrix Elements The one-electron contributions to the Fock matrix are R Fμ HF dr1 * μ (r ) 1 ( 1 2 ˆ dr1 * h μ = 2 1 ) = Tμ kinetic energy terms + k =1 NN ( r1 ) n I =1 dr1 * ˆ (2 J k μ dr1 * μ (r ) 1 ZI rI 1 ˆ Kk ) ( r1 ) n = Vμ ucl nuclear attraction terms Since there are Nbf basis functions in the basis set, Nbf 2 one-electron integrals of each type must be computed to construct the one electron matrix. Actually, since the kinetic and nuclear attraction operators are hermitian, only Nbf(Nbf+1)/2 integrals must be computed. Using Basis Sets to Solve the Hartree-Fock Equations Coulomb Matrix Elements The two-electron Coulomb contribution to the Fock matrix is R Fμ HF 2 n k =1 ˆ dr1 * ( r1 ) J k ( r1 ) μ = ˆ dr1 * h μ ( r1 ) = 2 =2 = = n k =1 n k =1 K + dr1 K 2 K =1 =1 * μ (r ) 1 n k =1 D dr2 K * c kc * k ( r2 ) k ( r2 ) * K dr1dr2 c* k c dr1dr2 * μ (r ) 1 ( r2 ) dr2 k ˆ Kk ) ( r1 ) r12 =1 =1 k * ˆ (2 J k μ dr1 k =1 dr1 * ( r1 ) μ =1 =1 K n ( r2 ) ( r1 ) r12 * μ (r ) 1 * ( r1 ) ( r2 ) ( r2 ) r12 ( r1 ) * r12 density matrix Using Basis Sets to Solve the Hartree-Fock Equations ( r2 ) ( r2 ) = K K =1 =1 D μ| Exchange Matrix Elements The two-electron exchange contribution to the Fock matrix is R Fμ HF n k =1 ˆ dr1 * ( r1 ) K k ( r1 ) μ = ˆ dr1 * h μ ( r1 ) = = = = n k =1 n k =1 K + dr1 * ( r1 ) μ dr1 n K * μ (r ) 1 K dr2 K =1 =1 1 2 D * dr1dr2 ( r1 ) r12 * K k ˆ Kk ) * ˆ k ( r2 ) p12 k ( r2 ) =1 =1 c kc * ˆ (2 J k μ dr1 k =1 =1 =1 k =1 K n c* k c dr1dr2 * μ (r ) 1 k dr2 ˆ ( r2 ) p12 ( r2 ) ( r1 ) r12 * μ (r ) 1 ( r1 ) * ( r2 ) ( r2 ) r12 ( r1 ) r12 * ( r2 ) ( r2 ) = 1 2 K K =1 =1 D μ| There are O(Nbf4/8) two-electron integrals, [μ | ]. The cost of computing these integrals dominates the cost of the integral calculation. Using Basis Sets to Solve the Hartree-Fock Equations Matrix Elements of the Fock Operator Combining the one-electron and the two-electron terms, the Fock matrix elements become R Fμ HF = Tμ + Vμnucl + Nbf Nbf D =1 =1 (μ | 1 2 μ| ) = Tμ + Vμnucl + Gμ So, the problem reduces to calculating the one-electron kinetic energy, nuclear attraction and two-electron 1/r12 integrals over the selected set of basis functions, { μ}, and then solving the matrix equation F RHF C = S C Solution of this equation involves transformation to an orthonormal basis set (the { μ} are usually normalized but not orthogonal) and then diagonalization of the resulting matrix. The eigenvalues of the matrix correspond to the orbital energies, { ii}, and the eigenvectors, transformed back to the original basis set, correspond to the expansion coefficients, {c i}. Using Basis Sets to Solve the Hartree-Fock Equations RHF Energy The electronic contribution to the total RHF energy is RHF Ee =2 n i =1 =2 n i =1 h ii + n n i =1 j =1 (2J ij Dμ (Tμ + Vμ ) + K ij ) n n i =1 j =1 ( Dμ 2 μ | The nuclear contribution is EN = NN NN I =1 J = I ZI ZJ RIJ and the total energy is RHF RHF Etotal = Ee + E N Using Basis Sets to Solve the Hartree-Fock Equations μ| ) Self-Consistent Field (SCF) Procedure There is one complication with the solution of the matrix RHF equations, the Fock matrix elements depend on the solutions to the equations through the density matrix, D . To address this problem, the equations are solved iteratively: 1. 2. 3. Specify the molecule and geometry: nuclear coordinates, {RI}; nuclear charges, {ZI}; and number of electrons (Ne). Select the basis set, { μ}. Calculate the integrals: Tμ , Vμ nucl, [μ | ] and Sμ . 4. Construct a guess for the density matrix, D 5. 6. Construct the Fock matrix. Solve the matrix RHF equations, obtaining a new set of coefficients, {c 7. 8. Form a new density matrix D new. Determine if the difference between the new and old density, |D matrix is below a set threshold. If not, return to step #5. 9. Calculate the total energy of the molecule at the given geometry. old, or expansion coefficients, {c Using Basis Sets to Solve the Hartree-Fock Equations new – i D i new}. old|, old}. Selection of Basis Functions We will now consider the selection of the functions to be used in basis sets for solving the RHF equations for atoms and molecules. There are two important criteria for selecting these functions. 1. 2. The functions must be capable of providing an accurate description of the atomic and molecular RHF wave functions. It must be possible to rapidly calculate the integrals Tμ , Vμ nucl, [μ | ] and Sμ with the functions. In addition, we want the basis sets to be as compact as possible. Using Basis Sets to Solve the Hartree-Fock Equations Hydrogenic Orbitals and Slater Functions The solutions of the electronic Schrödinger equation for the hydrogen atom are: n=2 n=1 1s = Z3 e Zr 2s 1 Z3 = (2 Z r ) e 42 1 Zr 2 2 px 1 Z3 = Z xe 42 1 Zr 2 etc. There are natural functions to use in the solution of the electronic Schrödinger equation. In 1930, John C. Slater proposed a slight modification of the radial functions in the hydrogenic orbitals, namely r n 1e r Slater functions have been used extensively in calculations on atomic and diatomic molecules and, if the exponents ( ) are optimized, have been shown to yield very accurate solutions of the HF equations. Unfortunately, it is very difficult to compute the multi-center, two-electron integrals, [μ | ], with Slater functions. As a result, it is not feasible to use Slater functions in calculations on polyatomic molecules. Using Basis Sets to Solve the Hartree-Fock Equations Gaussian Functions In 1950, S. F. Boys noted that all of the integrals, including the troublesome multicenter, two-electron integrals could be computed analytically if Gaussian functions l nlm mn ( x , y , z ) = N nlm x y z e r2 were used instead of Slater functions (Nnlm is a normalization constant). The occurrence of r2 in the exponential, rather than r, causes Gaussian functions to be less appropriate than Slater functions for use in atomic and molecular calculations. 1. 2. Mathematically, it can be shown that the wave function should have a cusp at the nucleus (origin). The derivative of the Gaussian function is continuous (and zero). So, Gaussian functions will have difficulty describing the wave function near the nucleus. Mathematically, it can also be shown that, at large distances from the nucleus, the wave function should fall off exponentially with r. So, Gaussian functions will have difficulty describing the “tail” of the wave function. Using Basis Sets to Solve the Hartree-Fock Equations Gaussian Functions and the Hydrogen Atom To better understand how Gaussian functions may fare in atomic and molecular calculations, consider their use in the hydrogen atom. 1.0x10-1 B B 1.0x10-2 Error in Energy (h) B -3 1.0x10 Four Gaussian functions achieve millihartree accuracy B B -4 1.0x10 B B 1.0x10-5 B Ten Gaussian functions achieve microhartree accuracy B 1.0x10-6 B B -7 1.0x10 B B B -8 1.0x10 1.0x10-9 B 0 5 n 10 Not great, but not prohibitive! 15 Using Basis Sets to Solve the Hartree-Fock Equations Gaussian Functions and the Neon Atom Let us now examine the somewhat more difficult case of the neon atom. More functions are required to achieve comparable accuracy in this case. 1.0x100 B s-set B B J B J B B J B J B J -1 1.0x10 Error in Energy (h) -2 1.0x10 1.0x10-3 (10s)-(11s) and (6p) Gaussian functions achieve millihartree accuracy B B J B B J B J B B J B J B p-set B J B J B J -4 1.0x10 -5 1.0x10 1.0x10-6 1.0x10-7 (19s) and (11p) Gaussian functions achieve microhartree accuracy -8 1.0x10 0 5 10 n 15 20 25 Using Basis Sets to Solve the Hartree-Fock Equations Gaussian Basis Sets Much work has been done on determining the optimum Gaussian basis sets to use in molecular HF calculations. This has resulted in a proliferation of “recommended” basis sets. Many of these basis sets have been collected in the Gaussian Basis Set Library at Pacific Northwest National Laboratory: http://www.emsl.pnl.gov/forms/basisform.html In 2001, F. Jensen (Ref. 1) reported a systematic study of Gaussian basis sets for use in molecular HF calculations. These studies defined a sequence of basis sets that systematically approach the complete basis set limit. To provide an accurate solution of the molecular HF equations, the basis sets must first and foremost provide an accurate description of the atomic HF orbitals. But, when a molecule is formed from atoms, the atomic orbitals are perturbed. The orbitals may contract or expand. They may become polarized toward or away from the neighboring atoms. They may even delocalize onto nearby atoms. The basis set must be able to describe all of these effects. Using Basis Sets to Solve the Hartree-Fock Equations Augmentation of Gaussian Basis Sets Contraction/Expansion of the Atomic Orbitals If several Gaussian functions are used to describe an atomic orbital, most of the contraction or expansion of the orbital can be accounted for by simply varying the coefficients in the basis function expansion. Thus, expansion and contraction can be accounted for except for the smallest atomic basis sets. Polarization of the Atomic Orbitals To describe polarization effects, additional functions must be added to the basis set. We will discuss this topic next. Delocalization of the Orbitals Delocalization of the atomic orbitals is equivalent to building ionic character into the wave function. If there is extensive delocalization, extra functions may be needed to describe the ionic character of the wave function. We will discuss this topic after we discuss polarization functions. Using Basis Sets to Solve the Hartree-Fock Equations Polarization Functions for Gaussian Basis Sets When a molecule like H2 is formed, each of the atomic orbitals, in this case a 1s-orbital at R = ∞, polarizes toward the other atom to help strengthen the bond. It does this by mixing 2p character into the wave function. + 1s + – + 2p = 1s + 2p So, polarizing the atomic orbitals requires the addition of higher angular momentum functions to the basis set. Jensen (Ref. 1) also systematically investigated the polarization functions to include in the basis sets. He did this through calculations on a set of prototypical molecules. The (sp) and polarization sets (dfg…) were matched by requiring that the sets give rise to identical errors (to the extent that it was possible to do so). Using Basis Sets to Solve the Hartree-Fock Equations Energy Lowerings in HF Calculations on N2 1x100 J J -1 1x10 H -3 1x10 H Ñ Ñ -6 1x10 B F B B B J 1x10 B B J 0 5 h i (18s11p6d3f2g1h) B J -7 1x10-8 É Ç B J H Ç g B B J H F (13s8p4d2f1g) B J H É f Ñ (10s6p2d1f) d B J H F 1x10-5 B J H 1x10-4 B J p F B J F J (7s4p1d) s H BB J 1x10-2 En,n-1 (h) B B 10 n 15 20 Using Basis Sets to Solve the Hartree-Fock Equations 25 Polarization Consistent Basis Sets Based on convergence studies of a number of prototypical molecules, Jensen proposed the following Gaussian basis sets for use in molecular calculations: Set H C–F pc-0 (3s) (5s3p) pc-1 (4s1p) (7s4p1d) pc-2 (6s2p1d) (10s6p2d1f) pc-3 (9s4p2d1f) (14s9p4d2f1g) pc-4 (11s6p3d2f1g) (18s11p6d3f2g1h) These sets are referred to as polarization consistent basis sets (pc-n). Using Basis Sets to Solve the Hartree-Fock Equations pc-n Sets: Oxygen s-Exponents core L L L L L L L L L LLL L L L L 2 L L L L L LLLLL L L L 1 L L 1x103 1x104 1x105 1x106 L L L L LL LL L LLL LL Using Basis Sets to Solve the Hartree-Fock Equations 1x10-1 L 1x100 L 1x101 L 1x102 L 3 1x107 pc-n 4 valence Convergence of NH Properties with pc-n Sets Re μe 1.01731 1.6133 5.51 0.55 0.007 0.000 0.0844 0.0636 0.0082 0.0006 De HF Limita 48.73 Errorb (7s4p1d) (10s6p2d1f) (14s9p4d2f1g) (18s11p6d3f2g1h) 2.82 0.54 0.038 0.003 a J. Kobus, private communication. b In kcal/mol, milliÅ, and Debyes. Using Basis Sets to Solve the Hartree-Fock Equations Convergence of CO Properties with pc-n Sets Re μe 1.10178 0.2650 4.09 0.46 0.067 0.014 0.0333 0.0244 0.0021 0.0003 De HF Limita 183.79 Errorb (7s4p1d) (10s6p2d1f) (14s9p4d2f1g) (18s11p6d3f2g1h) 4.71 0.852 0.060 0.0006 a J. Kobus, private communication. b In kcal/mol, milliÅ, and Debyes. Using Basis Sets to Solve the Hartree-Fock Equations Diffuse Functions for Gaussian Basis Sets In 2002, Jensen (Ref. 3) explored the addition of diffuse functions to the pc-n sets developed for density functional (DF) calculations. Because of the similarity between DF and HF basis sets, these functions can also be used to supplement the HF basis sets also. Addition of diffuse functions to the pc-n basis sets were shown to significantly improve the convergence of a number of molecular properties such as the dipole and quadrupole moments and dipole polarizabilities as well as electron affinities. The largest improvements were realized by adding a single diffuse function to the lower angular momentum sets (s, p, and d). Using Basis Sets to Solve the Hartree-Fock Equations Contraction of Gaussian Basis Sets Many of the steps in HF calculations depend on the number of functions in the basis set. The size of the basis set can be reduced by combining the primitive basis functions ( pprim) into contracted basis functions ( pcont): cont μ = L p =1 prim d pμ p Although this does not reduce the amount of work required to compute the integrals, it does reduce the work needed to diagonalize the Fock matrix. One way to contract the basis set is to take the expansion coefficient, {dpμ}, from the atomic orbitals. This defines a minimum basis set of contracted functions. Unfortunately, such a restricted set of contracted functions cannot describe the impact of molecular formation on the atomic orbitals. A very effective way to include these effects is to add the most diffuse primitive functions in the basis set to the minimal basis set. These are the functions that overlap the nearby atoms and provide the flexibility needed to describe the perturbations due to molecular formation. Using Basis Sets to Solve the Hartree-Fock Equations Contracted pc-2 Set for Oxygen Atom: [4s3p] 1(1s) s [4s] 14989.10 2248.08 511.616 144.713 46.9056 16.6009 6.16341 1.72634 0.679615 0.236456 p [3p] 62.0205 14.3226 4.35340 1.49260 0.516428 0.170182 0.000522 0.004038 0.020725 0.080526 0.232576 0.433056 0.347467 0.041790 -0.008217 0.002398 1(2p) 0.006269 0.043187 0.165515 0.361422 0.447563 0.235753 2(2s) -0.000118 -0.000920 -0.004746 -0.019065 -0.058999 -0.137038 -0.174532 0.167974 0.603102 0.372481 3 4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 2 3 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 Using Basis Sets to Solve the Hartree-Fock Equations Contracted pc-2 Set for Oxygen Atom: [2d1f] d [2d] 1 2.3 0.65 1.0 0.0 f [1f] 0.0 1.0 1 1.2 2 1.0 The above basis set is referred to as a generally contracted basis set because a primitive function can appear in any number of contracted functions, e.g., the primitive s functions contribute to at least two contracted functions, 1 and 2. If a primitive function appears in only one contracted function, the set is referred to as a segmented contracted basis set. Generally contracted sets are more efficient that segmented contracted sets for molecular calculations. Nonetheless, most of the sets in the literature are segmented sets because most integral programs cannot efficiently handle generally contracted sets. Using Basis Sets to Solve the Hartree-Fock Equations Contracted pc-n Gaussian Basis Sets Based on studies of a number of molecules, Jensen (Ref. 1) proposed the following contracted Gaussian basis sets for use in molecular calculations: Set H C–F pc-0 [2s] [3s2p] pc-1 [2s1p] [3s2p1d] pc-2 [3s2p1d] [4s3p2d1f] pc-3 [5s4p2d1f] [6s5p4d2f1g] pc-4 [7s6p3d2f1g] [8s7p6d3f2g1h] Using Basis Sets to Solve the Hartree-Fock Equations Convergence of NH Properties with pc-n Sets Re μe 1.01731 1.6133 De HF Limita 48.73 Errorb [3s2p1d] [4s3p2d1f] [6s5p4d2f1g] [8s7p6d3f2g1h] 3.77 0.612 0.046 0.003 10.78 0.58 0.008 0.000 a J. Kobus, private communication. b In kcal/mol, milliÅ, and Debyes. Using Basis Sets to Solve the Hartree-Fock Equations 0.0560 0.0651 0.0082 0.0006 References 1. “Polarization consistent basis sets: Principles,” F. Jensen, J. Chem. Phys. 115, 9113-9125 (2001). 2. “Polarization consistent basis sets. II. Estimating the Kohn-Sham basis set limit,” F. Jensen, J. Chem. Phys. 116, 7372-7379 (2002), 3. “Polarization consistent basis sets. III. The importance of diffuse functions,” F. Jensen, J. Chem. Phys. 117, 9234-9240 (2002). 4. “Polarization consistent basis sets. IV. The basis set convergence of equilibrium geometries, harmonic vibrational frequencies, and intensities,” F. Jensen, J. Chem. Phys. 118, 2459-2463 (2003). 5. “Polarization consistent basis sets. V. The elements Si-Cl,” F. Jensen, J. Chem. Phys. 121, 3463-3470 (2004). Using Basis Sets to Solve the Hartree-Fock Equations ...
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This note was uploaded on 12/07/2011 for the course CHEM 350 taught by Professor Duanejohnson during the Summer '06 term at University of Illinois, Urbana Champaign.

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