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Unformatted text preview: 5 MANY-ELECTRON QUANTUM MECHANICS 5.1 Introduction: Up to this stage we have not yet explicitly treated any quantum mechanical problem involving more than one electron. In this chapter we extend the rules of quantum mechanics to correctly construct many-electron wave functions, respecting the fact that electrons are indistinguishable particles. This turns out to be fundamentally related to the spin of the electron- we would be making very different wavefunctions if the spin of an electron was not one half. Section 5.2 treats the problem of electron spin followed in Section 5.3 by the consequences of indistinguishability (that consequence is already known to you as the Pauli Principle). In general, the quantum mechanics of many-electron systems cannot be solved exactly. Section 5.4 discusses the first fundamental approximation to make things tractable: the introduction of a basis set, usually of atomic-like orbitals (see the H atom in Chapter 3). The basis set approximation converts the many-electron problem from a partial differential equation into algebraic equations which are more suitable for computer solution. A vast array of basis sets of different qualities are available and you will be using many of them later on in practical calculations. An initial introduction is provided in Sec. , and then in Sec. 4 we discuss the exact solution of the many-electron problem in a given basis set. This exact solution, called full configuration interaction (FCI), is feasible for very small numbers of electrons, where it provides a benchmark against which other approximations can be tested. The rest of the book is about making and using those other approximations. Can one say anything in advance about what properties of FCI should be retained and which can be discarded? The final section of this chapter discusses this issue. The outcome is a set of criteria that well-defined approximations should have. 5.2 Electron spin The postulates or rules of QM discussed to this point (see Chapter 2) have made no mention of anything like spin. However, as shown in Figure XX below, electrons in fact behave as if they have spin magnetic moments. Passing a beam of H atoms through a magnetic field causes it to split into 2 components, as if there is a magnetic moment which can be either up (which is alpha spin;↑) or down (beta spin;↓). Remember the H atom is in its 1s level which means there is no magnetic moment due to orbital(l) angular momentum. Therefore if there is a magnetic moment, it must be due to the electron of the H atom. Well, actually it could also possibly be the proton, though in fact given the same angular momentum, any proton magnetic moment is expected by classical electromagnetism to be smaller by approximately the ratio of their masses since greater mass means smaller angular velocity and thus smaller magnetic moment. Figure XX.1: A beam of H atoms is split into 2 components upon passing through a strong magnetic field. This experiment was originally performed by Stern & Gerlach in the late 1920’s. To properly account for electron spin turns out to require not just quantum mechanics, but relativistic QM (the Dirac equation if you want to explore this further). In place of this, instead hypothesize that an electron has intrinsic spin angular momentum. s =1/ 2 (5.1) $ ms = + 1 2 (! spin or spin-up ) % & ms = " 1 2 (# spin or spin-down ) With spin ½, an electron will have intrinsic total angular momentum of magnitude 3! 2 , with a component in the z direction of magnitude ! 2 . A classical particle with charge e, mass me and angular momentum ! will develop a magnetic moment e! (5.2) µ! = " = 2.274 # 10 "24 J T"1 2 me In fact the quantum electron develops a spin magnetic moment µ = m s gs µ ! gs = gyromagnetic ratio = 2.00232 (5.3) 1 ms = ± 2 with a gyromagnetic ratio quite different to the value of 1 that we should expect based on directly applying our hypothesis (another intriguing consequence of relativistic quantum mechanics). Apart from this difference, this approach is a satisfactory way of dealing with the consequences of spin for quantum mechanics in chemistry. For the hydrogen atom, electron spin requires that we specify the state of the electron by the 4th quantum number, ms, in addition to the 3 spatial quantum numbers n, l, m , already discussed in Sec XX. The “spin eigenfunctions” are denoted as ! and ! and satisfy ˆ Sz ! = ! 2 ! (5.4) ˆ Sz " = # ! 2 " as well as the total spin angular momentum quantization condition: ˆ (5.5) S 2 ! = 1 2 3 2 !2 ! Spin-orbitals: The eigenfunctions of the H atom are therefore not just functions of space [ x, y, z ] , but are also functions of a spin variable, ! . We shall introduce the term “spin-orbital” for the product of a spatial function such as a hydrogenic atomic () orbital, !n , l , m ( x, y, z ) , with a spin function ! ms (" ) , which will be either ! or ! . The spin-orbital is thus: n, l, m, ms = !n , l , m ( x, y, z ) " # ms ($ ) (5.6) Forming matrix elements between spin-orbitals means doing integrations over both space and spin variables. For instance, an overlap integral will be: "n ! , l ! , m ! ( x, y, z )# ms! ($ ) "n , l , m ( x, y, z )# ms ($ ) ! ! = & dx dy dz " !%, l ! , m ! ( x, y, z ) "nlm ( x, y, z ) & d $# ms!# ms ! n! (5.7) = ' n !n ' l !l' m !m' ms! ms The spin coordinate σ is written analogously to the spatial coordinates in the equation above, but it should not be thought of analogously. After all, since only 2 eigenstates are possible, it is natural to use 2 component column vectors for the eigenstates and 2x2 matrices for the electron spin operators- see exercise XXX). 5.3 Pauli principle for systems of identical particles Since electrons are indistinguishable particles, any observation or measurement on a multielectron system must be independent of how the electrons are labeled. If we ˆ denote as P the permutation operator that swaps the labels of 2 electrons, we expect that ˆ P should not alter the wavefunction. Or as an equation: ˆ P! "! (5.8) ˆ #P ! =$ ! ˆ In other words, we think ! should be an eigenfunction of P . This makes a lot of sense, because, after all the Hamiltonian doesn’t depend on how the individual particles are labeled, so we should expect that the permutation operator commutes with it: ˆˆ ! P, H # = 0 (5.9) " $ Indeed this is the case (see exercise XX). Furthermore, since commuting operators have common eigenfunctions, this leads us logically to the result in Eq. (5.8). We can ˆ therefore conclude that an energy eigenfunction (our goal) is also an eigenfunction of P . Some consequences of this fact are almost immediately apparent. First let’s ˆ ˆ consider the eigenvalues of P that are possible. Certainly acting with P twice will ˆˆ recover the original eigenfunction: PP ! = ! . So: !2 = 1 " ! = ±1 (5.10) ˆ Two fundamentally different symmetries are possible for the eigenfunctions of P , each implying its own statistics: ! = "1 # antisymetric # Fermi statistics ! = +1 # symmetric # Bose statistics Particles which obey Fermi statistics are called Fermions, while those whose wavefunctions are symmetric to label swaps are called Bosons. Electrons behave as Fermions, as do other elementary particles with half-integer spin. Elementary particles with integer spin behave as Bosons. From now on we focus on Fermions, since we’re interested in many-electron systems. Valid wavefunctions for Fermions must change sign when acted upon with the ˆ permutation operator, P , as one sees by using ! = "1 in Eq. (5.8). In other words, Fermionic wavefunctions are antisymmetric to pairwise permutations. The question that immediately arises is how do we make antisymmetric wavefunctions? For concreteness, consider a system of 2 electrons, labeled as 1 and 2. The simplest idea would be to describe each of them with a spin-orbital as introduced in Sec. 2.1. Thus we might write: ! (1, 2 ) = " A (1) " B ( 2 ) (5.11) Here ! A (1) means a function, A, that depends on the space (x,y,z) and spin coordinates of electron 1. We can test the symmetry or antisymmetry by permuting the labels 1 and 2: ˆ (5.12) P12 ! (1, 2 ) = " A ( 2 ) " B (1) The result is not symmetric (i.e. ! (1, 2 ) ), nor, in particular, is it antisymmetric (i.e. ! " (1, 2 ) ). Hence it is not a valid 2-electron wavefunction. Thus simple product wavefunctions must be generalized to obey either Fermion or Bose statistics. Maybe you can see what should be done instead. We should consider combining the original product function with its permuted version, with a minus sign for the latter: ! (1, 2 ) = " A (1) " B ( 2 ) # " A ( 2 ) " B (1) (5.13) By construction, this is antisymmetric: ˆ (5.14) P12 ! (1, 2 ) = " A ( 2 )" B (1) # " A (1)" B ( 2 ) = # ! (1, 2 ) If you are familiar with determinants from math classes, you can see that our improved, antisymmetric 2-electron wavefunction is actually a 2x2 determinant: " A (1) " B (1) (5.15) ! (1, 2 ) = "A (2) "B (2) A 2x2 determinant is thus the appropriate generalization of a simple product of 2 orbitals, such that it obeys Fermi statistics. If we are looking at a 3-electron system, then, correspondingly, a 3x3 determinant corresponds to an antisymmetrized product wavefunction: " A (1) " B (1) "C (1) ! (1, 2, 3) = " A ( 2 ) " B ( 2 ) "C ( 2 ) (5.16) " A ( 3) " B ( 3) "C ( 3) Determinants are expanded in terms of smaller determinants (co-factors), taking the sign r+c as ( !1) where r and c are the row and column positions, going across a row. Thus: ! (1, 2, 3) = " A (1) " B ( 2 ) "C ( 2 ) # " B (1) " A ( 2 ) "C ( 2 ) + "C (1) " A ( 2 )" B ( 2 ) (5.17) " B ( 3)"C ( 3) " A ( 3)"C ( 3) " A ( 3)" B ( 3) Each 2x2 determinant has 2 terms and so ! (1, 2, 3) has a total of 6 terms, corresponding to all possible permutations (3!) of the electron labels. One can generalize further, and an n-electron determinant has n! terms, corresponding to all possible permutations of n labels: " A (1) " B (1) ! " N (1) != 1 "A (2) "B (2) ! "N (2) n! " (5.18) "A (n) "B (n) ! "N (n) 1 Note we have inserted a normalization constant ( ) in front of the antisymmetrized n! product so that if the φ are orthonormal, then the full wavefunction is automatically normalized (you will verify this in exercise xx). Let’s conclude this section with a few comments: (1) We’ve argued that an n x n determinant is the simplest wavefunction for n electrons. It is a product that is adapted to be antisymmetric, and is the closest that quantum mechanics can let us come to separating the coordinates of each electron from each other. (2) A determinant is usually not an exact n-electron wavefunction. We’ll soon (next section) see what would be exact, but meanwhile think of the variational principle: if 2 determinants are combined together, the energy will generally be lowered relative to 1 determinant. (3) There is a direct connection to the Pauli principle for many-electron atoms. An atomic spin-orbital has 4 quantum numbers (n,l,m,ms) and every occupied orbital must have unique quantum numbers. If it did not, then 2 identical spin-orbitals would be occupied which means the determinant would have 2 identical rows, which causes the wavefunction to vanish (see e.g. Eq. (5.17)). (4) The Aufbau Principle, of putting electrons into spin orbitals one at a time, or spatial orbitals 2 at a time, follows from the combination of the variational principle and the Pauli principle. 5.4 The finite basis set approximation Valid many-electron wave functions are determinants in which different spin-orbitals are occupied. The spin-orbitals, depending on position and spin of just 1 electron, define a 1-particle basis set. The determinants, depending on the position and spin of all n electrons in the molecule of interest, define an n-particle basis set. You may want to remind yourself about the fundamentals of basis set expansion in Sec. ( ) before reading on here. Such basis sets are usually infinite-dimensional, but to make useful computer-solvable approximations, we will truncate both expansions and make them finite. These 2 approximations will together control how close our approximate computer-based solution to the electronic Schrödinger equation is to the unattainable real thing. In practical calculations later on you will learn how to assess the consequences of those errors for yourself. There are evidently 2 key questions which must be answered before we have a workable computer-based simulation of many-electron molecules: (1) What function will comprise the finite 1-particle basis set? The remaining part of this section start to provide the answers. (2) Given a 1-particle basis set of N function, how many many-electron basis functions (determinants) will be used to describe the n-electron wavefunction? The answer must lie between the minimum (1) and the maximum (all possible: NCn). The latter is discussed in §Sec 4 below, and the former is discussed in the following chapter… Exact solution Size of n-particle basis set Figure XX Size of 1-particle basis set §5.3.2 Gaussian atomic orbital basis sets There are various types of 1-particle basis sets in common use. For molecules, the most widely used basis sets are composed of atomic-like orbitals, centered on each atom of the molecule. This makes intuitive sense because if our molecule was at dissociation, then these basis functions, wµ ( r ! RA ) would be ideal: we would only need those that are occupied by the Aufbau principle. In a bonded molecule the molecular spin-orbitals will be linear combinations of the atomic orbitals: ! i ( r ) = " wµ ( r ) C µ i µ Therefore it is possible (indeed likely) that atomic orbitals which are unoccupied in the free atom (like the 3rd 2p orbital of C) will become partly occupied in molecules. For this reason the number of atomic orbital basis functions must generally exceed the number of electrons on the free atom. In order to have a well-defined number of basis functions per atom, it therefore seems that this number should be based only on the principal and angular momentum quantum numbers (n & l). All members of a shell (ml and ms) should be included. Thus the smallest possible (minimal) AO basis will include 1 basis function for each value of n, l that is fully or partly occupied in the free atom. Larger basis sets (see next sub-section) may allow more functions or higher l. In the minimal basis, H has 2 AO spin-orbitals (n=1, l=0, ml=0, ms= + ! 1 2 ) or 1 spatial orbital (ls-like function) while Li ! Ne have 5 spatial AO’s or 10 spin-orbitals. What precisely is the form of these atomic orbitals? It would be reasonable to take hydrogenic functions: nl wµ ( r - A) = ! Ylm ("# ) Rn ( rA ) exp ( $%rA ) where rA = r ! A if A is the atomic center and θ and φ are the angles in local polar coordinates. However, it turns out to be much more efficient for computer evaluation to use Gaussian-basis functions K Stu ! µ ( r " A) = #x A yA zA $ Dk exp & "% k rA2 ( ' ) k =1 Gaussian basis functions are grouped into shells of common l = s + t + u . For instance a p=shell (see table below) has 3 functions (x,y,z) each of which consists of the same fixed mixture of 1< different Gaussian functions, each with its own exponent. K is called the degree of contraction, and the coefficients Dk are called contraction coefficients. Shell ncartessian npure s 1 1 p 3 3 (sp) 4 4 d 6 5 f 10 7 Table XX: Different types of shells in commin use and the number of basis functions they contain in their native Cartesian representation and in the pure (spherical harmonic) representation. The reason for making contracted Gaussian AO’s is that they can better mimic the behaviour of hydrogenic exponential AO’s. For example, modeling the Is atomic orbital as a contraction of 3 Gaussians (i.e. K=3) gives a basis function that is quite faithful to the original (see Figure below). On the other hand, a single Gaussian function decays too fast at large ra and does not have the correct cusp at the nucleus as rA ! 0 . Fig. XX: Comparison of a hydrogen 1s orbital with least squares fits by 1 and 3 Gaussian functions. rA STO-3G basis: A minimal basis in which exponential atomic-like orbitals (also called Slater-type orbitals (STO) for J.C. Slater, who pioneered their use) are represented as a contraction of 3 Gaussian functions. The STO-3G basis is the smallest and thus least expensive atomic orbital basis set that is widely used. 3-21G and 6-31G split valence basis sets: The fact that each occupied atomic orbital is represented by only a single basis function in STO-3G means that molecular spin-orbitals (Eq. XX) are made of atomic contributions of fixed size. This constraint can be lifted by allowing 2 basis functions per valence atomic orbital: one inner function (more compact with respect to the nucleus) and one outer function (more extended). Variationally, with this type of split valence basis, the atomic contributions to molecular orbitals can either expand or contract with respect to the free atom (e.g. accompanying unequal sharing of charge in a bond). With p-type basis functions, expansion or contraction can also be different along different axial directions (x or y vs z). These qualitative improvements translate into quantitatively better (lower) energies via the variational principle. The 3-21G and 6-31G basis sets are split-valence sets where the notation indicates the number of Gaussians contracted together to represent core, inner and outer calence functions respectively. Thus in the 3-21G basis for C, the Is-like basis function is a contraction of 3 Gaussians, the inner 2s and 2p basis functions are single Gaussians. A total of 9 basis functions are thus assigned to each C atom. There would likewise be 2 per H and 13 per Si. Polarization functions: If an atom interacts with an ion, its atomic orbitals will be slightly polarized: in other words they will distort slightly either towards the ion (if it is positive) or away from it. Similar effects can and do occur in polar covalent bond formation. In a basis expansion centered on atoms, a slight shift of an atomic orbital off its center can be achieved by mixing with a small amount of a function of the next higher angular momentum as illustrated below in Fig XX. To achieve this effect on H atoms requires addition of a set of p function, while on C, the addition of a set of d functions is needed. Fig.XX: Schematic illustration of the way in which polarization effects (slight off-center shift of an orbital) can be modeled using atom-centered orbitals of the next higher angular momentum. - = -+ +- = (Can’t figure out how to get the ε in there…) The split valence 6-31G basis is frequently augment with d function on non-H atoms; defining the 6-31G(d) (or 6-31G*) basis. By convention 6-31G(d) uses Cartesian d functions (see Table XX) so on C there are 15 basis functions. If p-type polarization functions are also included on H, this defines the 6-31G(d,p) (or, equivalently, the 631G**) basis set. There are innumerable alternatives to the small number of basis sets described to this point. In addition, one can also extend the “splitting” of the valence to 3-way as in the 6-311G basis, and extend the polarization to include multiple sets and/or higher angular momentum. Thus augmentations such as (2d) or (3d) or as large as (3df) or (3d2f) have been formulated. The number of functions on C and H for a variety of the basis sets discussed are summarized in Table XX. Still larger basis sets are described late in Sec Y.Y.Y. Diffuse Functions: The changes of atomic orbital size embodied in split valence basis sets such as 6-31G or even 6-311G are inadequate when an entire electron is added to a small molecule to make an anion. To describe anious, or other problems like intermolecular interactions which are sensitive to the large-r tails of orbitals, it is necessary to add an extra set of diffuse valence basis functions to atoms. The 6-31+G basis adds 4 extra functions per C atom for this purpose, while 6-31++G also includes 1 extra function per H. Summary: The basis sets described here form the first part of a systematic approach towards the complete basis set limit. They are summarized in Table XX below, which also includes an estimate of their relative computational cost, which will scale with at least the 3rd or 4th power of the relative number of functions. While the relative costs rise steeply with the number of basis functions, good results for many molecular properties can be obtained even with the smaller basis sets. These issues will be explored extensively in Chapters XX, XX and XX. Basis name STO-3G 3-21G 6-31G(d) 6-31G(d,p) 6-311G(2df,p) 6-311++G(3df,2pd) Nc 5 9 15 15 30 39 NH 1 2 2 5 6 15 relative cost* 1 6-10 20-60 35-100 200-1000 700-6000 Table XX: Numbers of functions on C and H for different basis sets and estimates of computational effort relative to STO-3G. Those estimates assume N3 (lower) and N4 (upper) scaling and a 1:1 hydrogen to non-hydrogen ratio. 5.5 Theoretical model chemistries In the previous section, different viable finite l-particle basis sets were described. Once a molecule is chosen and a basis set is selected, the number of basis functions is determined. For instance the water molecule with the 6-31G basis has 13 spatial basis functions. It is then possible to see how many different possible many-electron basis functions can be generated. A given n-electron basis function is a determinant in which n of the N 1-electron functions are occupied, as we saw in Sec. XX. If n and N refer to the number of electrons and spatial basis functions then the number of determinants is (remembering the number of spin-orbital basis functions in 2N): (2 N )! L = 2 n Cn = n!(2 N ! n )! However, many of these determinants will have zero contribution. Suppose, for instance that n = n! + n" (number of α and β spin electrons). We can ignore all determinants with the wrong number of α and β electrons so that: ( N !)2 L = N Cn! • N Cn" = n! !n" !( N # n! )!( N # n" )! Many of these may still give zero contribution (for instance due to symmetry of the molecule) but that will differ case-by-case. Given the complete set of configurations { !i i = 1… L } the exact solution to the Schrödinger equation in the finite basis follows from the linear variational principle (Sec. XX). The exact wavefunction, called the Full Configuration Interaction (FCI) wavefunction is: L ! = # "i ci i =1 where the expansion coefficients {ci , i = 1… L } are the result of diagonalizing the manyelectron Hamiltonian matrix, in the basis H c = Ec !! { ! } , for the lowest eigenvalue, E. i ! ˆ H ij = "i H " j We have assumed (without loss of generality) that the AO basis was orthonormalized so that !i ! j = " ij . So, once a 1-particle basis is specified, FCI gives the exact energy of the ground state, as long as we allow all L determinants to mix together. By the variational principle, any truncated configuration interaction which leaves out a non-zero contribution must give a higher energy. The problem is that L rises very fast with the size of the molecule, as shown in Table XX below for alkanes in the smallest possible AO basis (the minimal STO-3G basis). Since the effort to find the vector {ci , i = 1… L } and the ground state energy, E, must be at least proportional to L (there are L different ci, after all), it will generally be unfeasible. molecule CH4 C2H6 C3H8 C4H10 nα=nβ 5 9 13 17 Nsto-3G 9 16 23 30 L 1•6x104 1•3x108 1•3x1012 1•4x1016 In the example of the alkanes in the STO-3G basis, lengthening the chain by 1 CH2 unit increases the length of the expansion by 10,000 times or so. Thus computer time also increases by at least this factor when lengthening the chain by 1 CH2 unit. This type of dependence on system size is essentially exponential because t α L≈ (10,000)m for CmH2m+2. To make a practical computer-based model that approximates the Schrödinger equation and can be applied to a wide range of molecular sizes, we will require that the computational effort scales only algebraically with size of the molecule, m, rather than exponentially. The number of basis functions, N, for instance Nsto-3g=7m+2 for the alkanes in Table XX above, so a method that scales N1 or N2 or N3 would be fine. The models developed in Chapters XX & XX below meet this criterion. 5.4.1 Criteria for a good theoretical model chemistry The presentation of FCI above shows that even with small AO basis sets, it will only be feasible for the tiniest molecules. Hence to ensure feasibility, an approximation to the Schrödinger equation involves (i) A 1-particle basis truncation: the finite AO basis (ii) An n-particle basis truncation: this will define how accurately we model the interactions (or correlations) between electrons. With specification of the molecule (the atoms and their positions, numbers of α and β electrons), the AO basis, and the correlation treatment, a computer program can then produce the approximate energy, E. A wide variety of criteria can be used to assess the performance of the computational model that results. We collect a list of such criteria below. We first note that while some of the conditions do not have simple yes/no answers to them (for instance accuracy must be assessed across a wide range of chemical problems as is addressed in Chs. XX, XX…) other are yes/no conditions and can be used to exclude wide classes of possible models (for instance the size consistency condition will rule out truncated configuration interaction models). (1) Accuracy- a theoretical model must be capable of sufficient accuracy to be of predictive use. (2) Efficiency- the computational cost must not scale so steeply with molecular size as to prevent practical application to systems of interest. (3) Definition- the model should be well-defined in the sense of requiring no molecular-specific input apart from atomic positions, numbers of α and β electrons and spin state. Some models may violate this criterion by, for instance, specifying a Lewis structure which is only valid around one isomer. (4) Potential surface continuity- the model should yield an energy E ({ R! }) which is continuous as the nuclear positions { Ra } are varied. Failure to respect this criterion makes geometry optimization and the search for reaction paths unfeasible. Furthermore, Newtonian molecular dynamics only conserves energy on continuous potential energy surfaces. (5) Size-consistency- in order to allow meaningful calculations of reaction energies, it is essential that the quality of the model should not degrade as the molecule gets larger, so long as the nature of the bonding does not get more complex. For instance, if one H2 molecule is correct, then 2 H2 molecules far apart should also be correct. This property is termed size-consistency and means that E ( A! B ) = E ( A ) + E ( B ) where the energies are due to separate calculations on A alone, B alone and A and B together but far enough separated that they do not interact. (6) Variationality- To judge the quality of absolute energies it is helpful if they obey the variational principle (see Sec. X.XX). In practice this property is quite often sacrificed in order to achieve better accuracy and/or efficiency as we shall see later (e.g. density functional theory, Ch. XX) ...
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