Hartree-Fock-Roothaan wavefunctions

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

S. D. Peyerimhoff,/. Chem. Phys., 43, 998 (1965). Hartree—Fock Roothaan Wavefunctions,... 

Clementi, E. and Roetti, C., Roothaan-Hartree-Fock Atomic Wavefunctions, Atomic Data and Nuclear Data Tables 14, 177 (1974). 

These O, are called Linear Combination of Atomic Orbitals Molecular Orbitals (LCAO MOs) and if they are introduced into the Hartree-Fock equations (eqns (10-2.5)), a simple set of equations (the Hartree-Fock-Roothaan equations) is obtained which can be used to determine the optimum coefficients Cti. For those systems where the space part of each MO is doubly occupied, i.e. there are two electrons in each 0, with spin a and spin respectively so that the complete MOs including spin are different, the total wavefunction is... 

Our approximations so far (the orbital approximation, LCAO MO approximation, 77-electron approximation) have led us to a tt-electronic wavefunction composed of LCAO MOs which, in turn, are composed of 77-electron atomic orbitals. We still, however, have to solve the Hartree-Fock-Roothaan equations in order to find the orbital energies and coefficients in the MOs and this requires the calculation of integrals like (cf. eqns (10-3.3)) ... 

This restriction is not demanded. It is a simple way to satisfy the Pauli exclusion principle, but it is not the only means for doing so. In an unrestricted wavefunction, the spin-up electron and its spin-down partner do not have the same spatial description. The Hartree-Fock-Roothaan procedure is slightly modified to handle this case by creating a set of equations for the a electrons and another set for the p electrons, and then an algorithm similar to that described above is implemented. 

In order to solve for the energy and wavefunction within the Hartree-Fock-Roothaan procedure, the AOs must be specified. If the set of AOs is infinite, then the variational principle tells us that we will obtain the lowest possible energy within the HF-SCF method. This is called the HF limit, This is not the actual energy of the molecule recall that the HF method neglects instantaneous electron-electron interactions, otherwise known as electron correlation. 

S. Huzinaga and M, Klobukowski, J. Mol. Struct. (Theochem), 167, 1 (1988). Well-Tempered Gaussian Basis Set Expansions of Roothaan-Hartree-Fock Atomic Wavefunctions for Lithium Through Mercury. 

When the Schrodinger equation is solved in the Hartree-Fock— Roothaan procedure, the coefficients c,> are obtained and the wavefunction is at hand.2 Unfortunately, all the chemical information is contained in this wave-function, and it is expressed as a (very) long list of coefficients. As an example, a restricted Hartree-Fock calculation of benzene using the 6-31G basis set will have 102 atomic orbitals and 21 doubly occupied MOs for a total of 2142 coefficients. For the chemist, the interesting and pertinent data are entangled in a series of numbers, and the question becomes how to extract the chemical concepts from these numbers. 

Kim has formulated a relativistic Hartree-Fock-Roothaan equation for the ground states of closed-shell atoms using Slater-type orbitals. Relativistic effects in atoms have been reviewed by Grant. Malli and coworkers have formulated a relativistic SCF method for molecules. In this method, four-component spinor wavefunctions are obtained variationally in a self-consistent scheme using Gaussian basis sets. 

A Hartree-Fock calculation produces a set of molecular orbitals, known as the canonical Hartree-Fock orbitals. These orbitals are the solution to the Hartree-Fock-Roothaan equations and, while they may be used as the basis for a Cl expansion of the exact wave-function, they are not optimal in the sense that a different choice of orbital basis may result in a Cl expansion that converges more rapidly to the FCI limit. In 1955, Ldwdin demonstrated that the optimal one-electron basis for the Cl expansion of the exact wavefunction is the natural orbital basis [18]. In order to obtain the natural orbitals, we must first construct the first order reduced density matrix (RDM), defined as ... 

LCAO Approximation. Linear Combination of Atomic Orbitals approximation. Approximates the unknown Hartree-Fock Wavefunctions (Molecular Orbitals) by linear combinations of atom-centered functions (Atomic Orbitals) and leads to the Roothaan-Hall Equations. 

Roothaan-Hall Equations. The set of equations describing the best Hartree-Fock or Single-Determinant Wavefunction within the LCAO Approximation. 

Clementi E, Roetti C (1974) Tables of Roothaan-Hartree-Fock wavefunctions, special issue in atomic data and nuclear data table. Academic Press, New York... 

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... 

To assign values to the molecular orbital coefficients, c, many computational methods apply Hartree-Fock theory (which is based on the variational method).44 This uses the result that the calculated energy of a system with an approximate, normalized, antisymmetric wavefunction will be higher than the exact energy, so to obtain the optimal wavefunction (of the single determinant type), the coefficients c should be chosen such that they minimize the energy E, i.e., dEldc = 0. This leads to a set of equations to be solved for cMi known as the Roothaan-Hall equations. For the closed shell case, the equations are... 

The starting point for the vast majority of molecular wavefunctions is the Hartree-Fock framework as developed by Roothaan. The molecular wave-function F is expressed as an antisymmetrized product of molecular orbitals 0 each multiplied by its appropriate spin function,... 

Finite-field methods were first used to calculate dipole polarizabilities by Cohen and Roothaan [66]. For a fixed field strength V, the Hamiltonian potential energy term for the interaction between the electric field and ith electron is just The induced dipole moment with the applied field can be calculated from the Hartree-Fock wavefunction by integrating the dipole moment operator with the one-electron density since this satisfies the Hellmann-Feyman theorem. With the usual dipole moment expansion. 

The integral-driven procedure indicated above is practicable only if the elements of the two-particle density matrix can be rapidly accessed. In the closed-shell Hartree-Fock case, the two-particle density matrix can be easily constructed from the one-particle density. The situation is similar for open-shell and small multiconfigurational SCF wavefunctions the two-particle density matrix can be built up from a few compact matrices. In most open-shell Hartree Fock theories (Roothaan, 1960), the energy expression (Eq. (23))... 

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. 

---