Initial guess Hartree-Fock

As they stand, the Hartree-Fock equations 5.44, 5.46 or 5.47 are not very useful for molecular calculations, mainly because (1) they do not prescribe a mathematically viable procedure getting the initial guesses for the MO wavefunctions i ju which we need to initiate the iterative process (Section 5.2.3.5), and (2) the wavefunctions may be so complicated that they contribute nothing to a qualitative understanding of the electron distribution. 

Now, in the Hartree-Fock method (the Roothaan-Hall equations represent one implementation of the Hartree-Fock method) each electron moves in an average field due to all the other electrons (see the discussion in connection with Fig. 53, Section 5.23.2). As the c s are refined the MO wavefunctions improve and so this average field that each electron feels improves (since J and K, although not explicitly calculated (Section 5.2.3.63) improve with the i// s ). When the c s no longer change the field represented by this last set of c s is (practically) the same as that of the previous cycle, i.e. the two fields are consistent with one another, i.e. self-consistent . This Roothaan-Hall-Hartree-Fock iterative process (initial guess, first F, first-cycle c s, second F, second-cycle c s, third F, etc.) is therefore a self-consistent-field procedure or SCF procedure, like the Hartree-Fock procedure... 

Each Prs involves the sum over the occupied MO s (j = 1 -n we are dealing with a closed-shell ground-state molecule with 2n electrons) of the products of the coefficients of the basis functions 4>r and cf)s. As pointed out in Section 5.2.3.6.2 the Hartree-Fock procedure is usually started with an initial guess at the coefficients. We can use as our guess the extended Hiickel coefficients we obtained for HeH+, with this same geometry (Section 4.4.1.2) we need the c s only for the occupied MO s ... 

One guesses at an initial set of wave functions, , and constructs the Hartree-Fock Hamilton S which depends on the through the definitions of the Coulomb and exchange operators, (/ and One then calculates the new set of , and compares it (or the energy or the density matrix) to the input set (or to the energy or density matrix computed from the input set). This procedure is continued until the appropriate self-consistency is obtained. 

This initial guess may then be inserted on the right-hand sides of the equations and subsequently used to obtain new amplitudes. The process is continued until self-consistency is reached. For the special case in which canonical Hartree-Fock molecular orbitals are used, the Fock matrix is diagonal and the T2 amplitude approximation above is exactly the same as the first-order perturbed wave-function parameters derived from Moller-Plesset theory (cf. Eq. [212]). In that case, the Df and arrays contain the usual molecular orbital energies, and the initial guess for the T1 amplitudes vanishes. 

An important aspect of density-functional theory (like Hartree-Fock and other mean-field theories) is that the Schrodinger equation (9) has to be solved self-consistently, because the potentials Vn and V,c depend on the wavefunctions themselves. In its simplest form this means making an initial guess at the effective potential (7), then using the output wavefunctions and charge density to construct a new potential and iterating till convergence. 

The overall procedure to achieve self-consistency is very reminiscent of that used in Hartree-Fock theory, involving first an initial guess of the density by superimposing atomic densities, construction of the Kohn-Sham and overlap matrices, and diagonalisation to give the eigenfunctions and eigenvectors from which the Kohn-Sham orbitals can be... 

C is a square matrix of expansion coefficients, and is a veaor of the orbital energies. Equation [10] is solved in a manner similar to that used for the Hartree-Fock equations. That is, an initial guess of the coefficients is made, the Fock matrix is constructed, and then the Fock matrix is diagonalized to obtain the new coefficients and orbital energies. The new coefficients are then used to construrt a new Fock matrix, and the procedure is repeated until either the change in energy or orbital coefficients is below some threshold. 

Performing Kohn-Sham Density-Functional Calculations. How does one do a molecular density-functional calculation with (or some other functional) One starts with an initial guess for p, which is usually foimd by superposing calculated electron densities of the individual atoms at the chosen molecular geometry. From the initial guess for p(r), an initial estimate of u c( ) found from (15.127) and (15.131) and this initial v d ) is used in the Kohn-Sham equations (15.121), which are solved for the initial estimate of the KS orbitals. In solving 15.121), the flP s are usually expanded in terms of a set of basis functions Xr ( P = 2r=i to yield equations that resemble the Hartree-Fock-Roothaan equations (13.157) and (13.179), except that the Fock matrix elements = xr F x are replaced by the Kohn-Sham matrix elements = (Xr Xs), where is in (15.122) and(15.123).Thus, instead of (13.157), in KS DFT with a basis-set expansion of the orbitals, one solves the equations... 

On the other hand, the linear combination of atomic orbitals - molecular orbital (LCAO-MO) theory, is actually the same as Hartree-Fock theory. The basic idea of this theory is that a molecular orbital is made of a linear combination of atom-centered basis functions describing the atomic orbitals. The Hartree-Fock procedure simply determines the linear expansion coefficients of the linear combination. The variables in the Hartree-Fock equations are recursively defined, that is, they depend on themselves, so the equations are solved by an iterative procedure. In typical cases, the Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky cases, convergence may be improved by changing the form of the initial guess. Since the equations are solved self-consistently, Hartree-Fock is an example of a self-consistent field (SCF) method. 

Note that to solve Equation (12) for the orbitals igt we need to already know the orbitals to set up the coulomb and exchange potentials We get around this problem by doing an iterative solution of Equation (12) starting with an initial guess of the orbitals. After 10 or so iterations, we will get a self-consistent field (SCF) result. In the current jargon of quantum chemistry, SCF and Hartree-Fock are used interchangeably. 

The true Hamiltonian operator and wave function involve the coordinates of all n electrons. The Hartree-Fock Hamiltonian operator F is a one-electron operator (that is, it involves the coordinates of only one electron), and (14.25) is a one-electron differential equation. This has been indicated in (14.25) by writing F and , as functions of the coordinates of electron 1. Of course, the coordinates of any electron could have been used. The operator F is peculiar in that it depends on its own eigenfunctions [see Eqs. (14.26) to (14.29)], which are not known initially. Hence the Hartree-Fock equations must be solved by an iterative process, starting with an initial guess for the MOs. 

The best possible forms for the MOs 4>i of a molecule are the solutions of the Hartree-Fock equations = , where F is the Fock operator [Eqs. (14.26) to (14.29)] and s, is the orbital energy. To solve the Hartree-Fock equations, we expand (pi using a set of basis functions , = 2 c iXsl this leads to the Roothaan equations (14.34) for the coefficients c, and orbital energies s,. Since the Fock operator F and its matrix elements F [Eq. (14.41)] depend on the occupied MOs, which are unknown, the Roothaan equations are solved by an iterative process that starts with an initial guess for the occupied MOs. 

In a very recent study, Banerjee et al used different ab initio methods to study the properties of small Kjv clusters for even A with 2 < A < 20. They used both density-functional methods and post-Hartree-Fock approaches, where correlation is added either via Moller-Plesset perturbation theory or via the coupled-cluster approach (see, e.g., ref. 1). In order to determine the structures of the lowest total energy, they used as initial guesses structures from earlier studies on Nuat clusters that subsequently were allowed to relax to their closest total-energy minimum. Unfortunately, only the density-functional method was used in optimizing the structures, whereby a comparison between the two approaches is not made possible. 

In Table 11.3, the convergence of the inverse iterations (i.e. Newton s method) is illustrated by calculations on the water molecule at the CISD/cc-pVDZ level, carried out at the geometries of Table 11.2. At both geometries, the Hartree-Fock wave function is used as the initial guess. For each iteration, the error in the energy is listed. 

Second, compute the electron-attached Hartree-Fock determinant for the state of interest, using the neutral molecule s MOs as an initial guess but altering the initial occupancies such that an electron is added not to the LUMO but to some higher-lying virtual MO. 

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