Cramer C.J. Essentials of Computational Chemistry Theories and Models

where repeated application of Eq. (4.34) is used in proving that the energy eigenvalue of the many-electron wave function is simply the sum of the one-electron energy eigenvalues. Note that Eqs. (4.32)–(4.36) provide the mathematical rigor behind the Huckel¨ theory example presented more informally above. Note that if every ψ is normalized then HP is also normalized, since | HP|2 = |ψ1|2 |ψ2|2 · · · |ψN |2.

4.5.2The Hartree Hamiltonian

As noted above, however, the Hamiltonian defined by Eqs. (4.32) and (4.33) does not include interelectronic repulsion, computation of which is vexing because it depends not on one electron, but instead on all possible (simultaneous) pairwise interactions. We may ask, however, how useful is the Hartree-product wave function in computing energies from the correct Hamiltonian? That is, we wish to find orbitals ψ that minimize HP |H | HP . By applying variational calculus, one can show that each such orbital ψi is an eigenfunction of its own operator hi defined by

where the final term represents an interaction potential with all of the other electrons occupying orbitals {j } and may be computed as

where ρj is the charge (probability) density associated with electron j . The repulsive third term on the r.h.s. of Eq. (4.37) is thus exactly analogous to the attractive second term, except that nuclei are treated as point charges, while electrons, being treated as wave functions, have their charge spread out, so an integration over all space is necessary. Recall, however, that ρj = |ψj |2. Since the point of undertaking the calculation is to determine the individual ψ, how can they be used in the one-electron Hamiltonians before they are known?

To finesse this problem, Hartree (1928) proposed an iterative ‘self-consistent field’ (SCF) method. In the first step of the SCF process, one guesses the wave functions ψ for all of the occupied MOs (AOs in Hartree’s case, since he was working exclusively with atoms) and uses these to construct the necessary one-electron operators h. Solution of each differential Eq. (4.34) (in an atom, with its spherical symmetry, this is relatively straightforward, and Hartree was helped by his retired father who enjoyed the mathematical challenge afforded by such calculations) provides a new set of ψ, presumably different from the initial guess. So, the one-electron Hamiltonians are formed anew using these presumably more accurate ψ to determine each necessary ρ, and the process is repeated to obtain a still better set of ψ. At some point, the difference between a newly determined set and the immediately preceding set falls below some threshold criterion, and we refer to the final set of ψ as the ‘converged’ SCF orbitals. (An example of a threshold criterion might be that the total electronic energy change by no more than 10−6 a.u., and/or that the energy eigenvalue for each MO change by

are orthonormal and are typically denoted as α and β (not to be confused with the α and β of Huckel¨ theory!) The spin quantum number is a natural consequence of the application of relativistic quantum mechanics to the electron (i.e., accounting for Einstein’s theory of relativity in the equations of quantum mechanics), as first shown by Dirac. Another consequence of relativistic quantum mechanics is the so-called Pauli exclusion principle, which is usually stated as the assertion that no two electrons can be characterized by the same set of quantum numbers. Thus, in a given MO (which defines all electronic quantum numbers except spin) there are only two possible choices for the remaining quantum number, α or β, and thus only two electrons may be placed in any MO.

Knowing these aspects of quantum mechanics, if we were to construct a ground-state Hartree-product wave function for a system having two electrons of the same spin, say α, we would write

where the left superscript 3 indicates a triplet electronic state (two electrons spin parallel) and ψa and ψb are different from one another (since otherwise electrons 1 and 2 would have all identical quantum numbers) and orthonormal. However, the wave function defined by Eq. (4.40) is fundamentally flawed. The Pauli exclusion principle is an important mnemonic, but it actually derives from a feature of relativistic quantum field theory that has more general consequences, namely that electronic wave functions must change sign whenever the coordinates of two electrons are interchanged. Such a wave function is said to be ‘antisymmetric’. For notational purposes, we can define the permutation operator Pij as the operator that interchanges the coordinates of electrons i and j . Thus, we would write the Pauli principle for a system of N electrons as

where q now includes not only the three Cartesian coordinates but also the spin function. If we apply P12 to the Hartree-product wave function of Eq. (4.40),

we immediately see that it does not satisfy the Pauli principle. However, a slight modification to HP can be made that causes it to satisfy the constraints of Eq. (4.41), namely

1

3 SD = √ [ψa (1)α(1)ψb (2)α(2) − ψa (2)α(2)ψb (1)α(1)] (4.43)

2

(the reader is urged to verify that 3 SD does indeed satisfy the Pauli principle; for the ‘SD’ subscript, see next section). Note that if we integrate |3 SD|2 over all space we have

3 SD 2 dr1dω1dr2dω2 = 1 |ψa (1)|2 |α(1)|2 |ψb(2)|2 |α(2)|2dr1dω1dr2dω2

2

− 2 ψa (1)ψb(1) |α(1)|2 ψb(2)ψa (2) |α(2)|2 dr1dω1dr2dω2

+|ψa (2)|2 |α(2)|2 |ψb(1)|2 |α(1)|2dr1dω1dr2dω2

=1 (1 − 0 + 1)

2

where ω is a spin integration variable, the simplification of the various integrals on the r.h.s. proceeds from the orthonormality of the MOs and spin functions, and we see that the prefactor of 2−1/2 in Eq. (4.43) is required for normalization.

4.5.4 Slater Determinants

A different mathematical notation can be used for Eq. (4.43)

where the difference of MO products has been expressed as a determinant. Note that the permutation operator P applied to a determinant has the effect of interchanging two of the rows. It is a general property of a determinant that it changes sign when any two rows (or columns) are interchanged, and the utility of this feature for use in constructing antisymmetric wave functions was first exploited by Slater (1929). Thus, the ‘SD’ subscript used in Eqs. (4.43) –(4.45) stands for ‘Slater determinant’. On a term-by-term basis, Slaterdeterminantal wave functions quickly become rather tedious to write down, but determinantal notation allows them to be expressed reasonably compactly as, in general,

where N is the total number of electrons and χ is a spin-orbital, i.e., a product of a spatial orbital and an electron spin eigenfunction. A still more compact notation that finds widespread use is

and carry out the same evaluation of interelectronic repulsion we have

1

SD r12 SDdr1dω1dr2dω2

=1 (Jab + Jab )

2

Note that the disappearance of the exchange correlation derives from the orthogonality of the α and β spin functions, which causes the second integral in the second equality to be zero when integrated over either spin coordinate.

4.5.5 The Hartree-Fock Self-consistent Field Method

Fock first proposed the extension of Hartree’s SCF procedure to Slater determinantal wave functions. Just as with Hartree product orbitals, the HF MOs can be individually determined as eigenfunctions of a set of one-electron operators, but now the interaction of each electron with the static field of all of the other electrons (this being the basis of the SCF approximation) includes exchange effects on the Coulomb repulsion. Some years later, in a paper that was critical to the further development of practical computation, Roothaan described matrix algebraic equations that permitted HF calculations to be carried out using a basis set representation for the MOs (Roothaan 1951; for historical insights, see Zerner 2000). We will forego a formal derivation of all aspects of the HF equations, and simply present them in their typical form for closed-shell systems (i.e., all electrons spin-paired, two per occupied orbital) with wave functions represented as a single Slater determinant. This formalism is called ‘restricted Hartree-Fock’ (RHF); alternative formalisms are discussed in Chapter 6.

The one-electron Fock operator is defined for each electron i as

where the final term, the HF potential, is 2Ji − Ki , and the Ji and Ki operators are defined so as to compute the Jij and Kij integrals previously defined above. To determine the MOs using the Roothaan approach, we follow a procedure analogous to that previously described for Huckel¨ theory. First, given a set of N basis functions, we solve the secular equation

to find its various roots Ej . In this case, the values for the matrix elements F and S are computed explicitly.

Matrix elements S are the overlap matrix elements we have seen before. For a general matrix element Fµν (we here adopt a convention that basis functions are indexed by lowercase Greek letters, while MOs are indexed by lower-case Roman letters) we compute

The notation µ|g|ν where g is some operator which takes basis function φν as its argument, implies a so-called one-electron integral of the form

µ|g|ν = φµ(gφν )dr. (4.55)

Thus, for the first term in Eq. (4.54) g involves the Laplacian operator and for the second term g is the distance operator to a particular nucleus. The notation (µν|λσ ) also implies a specific integration, in this case

where φµ and φν represent the probability density of one electron and φλ and φσ the other. The exchange integrals (µλ|νσ ) are preceded by a factor of 1/2 because they are limited to electrons of the same spin while Coulomb interactions are present for any combination of spins.

The final sum in Eq. (4.54) weights the various so-called ‘four-index integrals’ by elements of the ‘density matrix’ P. This matrix in some sense describes the degree to which individual basis functions contribute to the many-electron wave function, and thus how energetically important the Coulomb and exchange integrals should be (i.e., if a basis function fails to contribute in a significant way to any occupied MO, clearly integrals involving that basis

where the coefficients aζ i specify the (normalized) contribution of basis function ζ to MO i and the factor of two appears because with RHF theory we are considering only singlet wave functions in which all orbitals are doubly occupied.

While the process of solving the HF secular determinant to find orbital energies and coefficients is quite analogous to that already described above for effective Hamiltonian methods, it is characterized by the same paradox present in the Hartree formalism. That is, we need to know the orbital coefficients to form the density matrix that is used in the Fock matrix elements, but the purpose of solving the secular equation is to determine those orbital coefficients. So, just as in the Hartree method, the HF method follows a SCF procedure, where first we guess the orbital coefficients (e.g., from an effective Hamiltonian method) and then we iterate to convergence. The full process is described schematically by the flow chart in Figure 4.3. The energy of the HF wavefunction can be computed in a fashion analogous to Eq. (4.39).

Hartree –Fock theory as constructed using the Roothaan approach is quite beautiful in the abstract. This is not to say, however, that it does not suffer from certain chemical and practical limitations. Its chief chemical limitation is the one-electron nature of the Fock operators. Other than exchange, all electron correlation is ignored. It is, of course, an interesting question to ask just how important such correlation is for various molecular properties, and we will examine that in some detail in following chapters.

Furthermore, from a practical standpoint, HF theory posed some very challenging technical problems to early computational chemists. One problem was choice of a basis set. The LCAO approach using hydrogenic orbitals remains attractive in principle; however, this basis set requires numerical solution of the four-index integrals appearing in the Fock matrix elements, and that is a very tedious process. Moreover, the number of four-index integrals is daunting. Since each index runs over the total number of basis functions, there are in principle N 4 total integrals to be evaluated, and this quartic scaling behavior with respect to basis-set size proves to be the bottleneck in HF theory applied to essentially any molecule.

Historically, two philosophies began to emerge at this stage with respect to how best to make further progress. The first philosophy might be summed up as follows: The HF equations are very powerful but still, after all, chemically flawed. Thus, other approximations that may be introduced to simplify their solution, and possibly at the same time improve their accuracy (by some sort of parameterization to reproduce key experimental quantities), are well justified. Many computational chemists continue to be guided by this philosophy today, and it underlies the motivation for so-called ‘semiempirical’ MO theories, which are discussed in detail in the next chapter.

The second philosophy essentially views HF theory as a stepping stone on the way to exact solution of the Schrodinger¨ equation. HF theory provides a very well defined energy, one which can be converged in the limit of an infinite basis set, and the difference between that

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Levine, I. N. 2000. Quantum Chemistry , 5th Edn., Prentice Hall: New York.

Lowry, T. H. and Richardson, K. S. 1981. Mechanism and Theory in Organic Chemistry , 2nd Edn., Harper & Row: New York, 82 – 112.

Szabo, A. and Ostlund, N. S. 1982. Modern Quantum Chemistry , Macmillan: New York.

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