The Hartree-Fock Method for Molecules

A key development in quantum chemistry has been the computation of accurate self-consistent-field wave functions for many diatomic and polyatomic molecules. The principles of molecular SCF calculations are essentially the same as for atomic SCF calculations (Section 11.1). We shall restrict ourselves to closed-shell configurations. For open shells, the formulas are more complicated. 

The molecular Hartree-Fock wave function is written as an antisymmetrized product (Slater determinant) of spin-orbitals, each spin-orbital being a product of a spatial orbital f i and a spin function (either a or jS). 

The expression for the Hartree-Fock molecular electronic energy hf is given by the variation theorem as hf = ( l ei + where D is the Slater-determinant 

Hartree-Fock wave function and H [ and are given by (13.5) and (13.6). Since F vv does not involve electronic coordinates and D is normalized, we have (DjV jvjvlD) = Vnn D D) - F/vv- Th operator is the sum of one-electron operators fi and two- 

Therefore, the Hartree-Fock energy of a diatomic or polyatomic molecule with only closed shells is 

The Hamiltonian H is the same as the Hamiltonian H for an atom except that 2aZ /replaces Z/r, in /,. Hence Eq. (11.83) can be used to give (F Ffei I Therefore, the Hartree-Fock energy of a diatomic or polyatomic molecule with only closed shells is 

The Hartree-Fock method looks for those orbitals that minimize the variational integral hf- Each MO is taken to be normalized ( , ( 1) , ( 1)) = 1. Moreover, for computational convenience one takes the MOs to be orthogonal ( ,( 1) 1)) = 0 for 

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Section 13.16 The Hartree-Fock Method for Molecules 429 Substitution of the expansion (13.156) into the Hartree-Fock equations (13.148)... 

We have seen the same in the Hartree-Fock method for molecules, where the Coulomb and exchange operators depended on the solutions to the Fock equation, (cf. p. 4121. 

Since the exact solution of the Hartree-Fock equation for molecules also proved to be impossible, numerical methods approximating the solution of the Schrodinger s equation at the HF limit have been developed. For example, in the Roothan-Hall SCF method, each SCF orbital is expressed in terms of a linear combination of fixed orbitals or basis sets ((Pi). These orbitals are fixed in the sense that they are not allowed to vary as the SCF calculation proceeds. From n basis functions, new SCF orbitals are generated by... 

A. C. Wahl, PJ. Bertoncini, G. Das, T.L. Gilbert, Recent progress beyond the Hartree-Fock method for diatomic molecules The method of optimized valence configurations, Int. J. Quantum Chem. S1 (1967) 123. 

The best possible wavefunction of the form of (10) is called the Hartree-Fock wavefunction. For molecules it is difficult to solve (11) numerically. The most widely used procedure was proposed by Roothaan.28 This involves expressing the molecular orbitals t/> (.x) as a linear combination of basis functions (normally atomic orbitals) and varying the coefficients in this expansion so as to find the best possible solutions to (11) within the limits of a given basis set. This procedure is called the self-consistent field (SCF) method. As the size and flexibility of the basis set is increased the SCF orbitals and energy approach the true Hartree-Fock ones. 

Determinantal MO s may be obtained by a large number of computational methods based on Roothaan s self-consistent field formalism 94> for solving the Hartree-Fock equation for molecules which differ in degree of sophistication as regards the completeness and kind of the set of starting atomic wave functions, as well as the completeness of the Hamiltonian used 9S>. So a chain of various kinds of approximations is available for calculations starting from different ways of non-empirical ab initio" calculations 96>, viasemiempiricalmethods for all-valence electrons with inclusion of electronic interaction 95-97>98)... 

In practice, using currently available exchange and correlation potentials, this path leads to results [113] worse than those obtained with the Hartree-Fock method. This is illustrated for momentum moments in Table 19.2 which shows median absolute percent errors of (p ) for 78 molecules relative to those computed by an approximate singles and doubles coupled-cluster method often called QCISD [114,115]. The molecules are mostly polyatomic, and contain H, C, N, O, and F atoms. The correlation-consistent cc-pVTZ basis set [110] was used for these computations. Table 19.2 shows the median errors for the Hartree-Fock method, for second-order Mpller-Plesset permrbation theory (MP2), and for DFT calculations done with the B3LYP hybrid density functional... 

A concrete example of the variational principle is provided by the Hartree-Fock approximation. This method asserts that the electrons can be treated independently, and that the -electron wavefimction of the atom or molecule can be written as a Slater determinant made up of orbitals. These orbitals are defined to be those which minimize the expectation value of the energy. Since the general mathematical form of these orbitals is not known (especially in molecules), then the resulting problem is highly nonlinear and formidably difficult to solve. However, as mentioned in subsection (A 1.1.3.2). a common approach is to assume that the orbitals can be written as linear combinations of one-electron basis functions. If the basis functions are fixed, then the optimization problem reduces to that of finding the best set of coefficients for each orbital. This tremendous simplification provided a revolutionary advance for the application of the Hartree-Fock method to molecules, and was originally proposed by Roothaan in 1951. A similar form of the trial function occurs when it is assumed that the exact (as opposed to Hartree-Fock) wavefimction can be written as a linear combination of Slater determinants (see equation (A 1.1.104 ) ). In the conceptually simpler latter case, the objective is to minimize an expression of the form... 

In general, the Hartree-Fock method indicates this SCF-based method. Despite the simplicity of the procedure, it soon became clear that solving this equation is not-trivial for usual molecular electronic systems. The Hartree-Fock equation essentially cannot be solved for molecules without computers. Actually, solving the Hartree-Fock equation for molecules had to await the appearance of general-purpose computers. 

ACES II Basis Sets Correlation Consistent Sets Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Integrals of Electron Repulsion Numerical Hartree-Fock Methods for Molecules Symmetry in Chemistry TURBOMOLE. 

Even though recent progress in hardware and software development has made it possible to study quite large molecules, systems of the size considered here do not lend themselves to studies with any ab initio technique. The Hartree-Fock method has the advantage of being size consistent, which Is a necessity for this type of study when results for molecules of vastly different size are to be compared. In addition, the method is technically and economically feasible for these large systems. 

The various methods used in quantum chemistry make it possible to compute equilibrium intermolecular distances, to describe intermolecular forces and chemical reactions too. The usual way to calculate these properties is based on the independent particle model this is the Hartree-Fock method. The expansion of one-electron wave-functions (molecular orbitals) in practice requires technical work on computers. It was believed for years and years that ab initio computations will become a routine task even for large molecules. In spite of the enormous increase and development in computer technique, however, this expectation has not been fulfilled. The treatment of large, extended molecular systems still needs special theoretical background. In other words, some approximations should be used in the methods which describe the properties of molecules of large size and/or interacting systems. The further approximations are to be chosen carefully this caution is especially important when going beyond the HF level. The inclusion of the electron correlation in the calculations in a convenient way is still one of the most significant tasks of quantum chemistry. 

In the Hartree-Fock method, the molecular (or atomic) electronic wave function is approximated by an antisymmetrized product (Slater determinant) of spin-orbitals each spin-orbital is the product of a spatial orbital and a spin function (a or ft). Solution of the Hartree-Fock equations (given below) yields the orbitals that minimize the variational integral. Thus the Hartree-Fock wave function is the best possible electronic wave function in which each electron is assigned to a spatial orbital. For a closed-subshell state of an -electron molecule, minimization... 

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