The SCF procedure

The system of equations (1.8) is based on the central field approximation, and therefore its application to real atoms is entirely dependent on the existence of closed shells, which restore spherical symmetry in each successive row of the periodic table. For spherically symmetric atoms with closed shells, the Hartree-Fock equations do not involve neglecting noncentral electrostatic interactions and are therefore said to apply exactly. This does not mean that they are expected to yield exact values for the experimental energies, but merely that they will apply better than for atoms which are not centrally symmetric. One should bear in mind that, in any real atom, there are many excited configurations, which mix in even with the ground state and which are not spherically symmetric. Even if one could include all of them in a Hartree-Fock multiconfigura-tional calculation, they would not be exactly represented. Consequently, there is no such thing as an exact solution for any many-electron atom, even under the most favourable assumptions of spherical symmetry. 

---

The original PCM method uses a cavity made of spherical regions around each atom. The isodensity PCM model (IPCM) uses a cavity that is defined by an isosurface of the electron density. This is defined iteratively by running SCF calculations with the cavity until a convergence is reached. The self-consistent isodensity PCM model (SCI-PCM) is similar to IPCM in theory, but different in implementation. SCI-PCM calculations embed the cavity calculation in the SCF procedure to account for coupling between the two parts of the calculation. 

SCF procedure is begun, and then used in each SCF iteration. Formally, in the large basis set limit the SCF procedure involves a computational effort which increases as the number of basis functions to the fourth power. Below it will be shown that the scaling may be substantially smaller in acmal calculations. 

The Multi-configuration Self-consistent Field (MCSCF) method can be considered as a Cl where not only the coefficients in front of the determinants are optimized by the variational principle, but also the MOs used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure (if the multi-configuration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number of configurations included, the size of MCSCF wave function that can be treated is somewhat smaller than for Cl methods. 

For computational purposes it is convenient to work with canonical MOs, i.e. those which make the matrix of Lagrange multipliers diagonal, and which are eigenfunctions of the Fock operator at convergence (eq, (3.41)). This corresponds to a specific choice of a unitary transformation of the occupied MOs. Once the SCF procedure has converged, however, we may chose other sets of orbitals by forming linear combinations of the canonical MOs. The total wave function, and thus all observable properties, are independent of such a rotation of the MOs. 

In this approach, the electron density of a solvated molecule (p) is calculated using the SCF procedure where the isolated molecule Hamiltonian Hgas is replaced by the solvated molecule Hamiltonian //sol ... 

There is a fundamental difference between Eqs. 4.12 and 4.15 despite their apparent similarity. The term electron density (see Eq. 4.13), whereas the term Vcxt in Eq. 4.12, is constant in the SCF procedure. To reflect this fact, the approach based on Eqs. 4.13-4.15 is frequently called the Self-Consistent Reaction Field method (SCRF). (Throughout the text, AXY/SCRF denotes combined quantum-mechanical/reaction field calculations where XXX specifies the quantum-mechanical method.)... 

The nonelectrostatic components of the free energy such as the energy of cavity formation AGcav or components that take into account atomistic details of the medium (interactions between atoms inside the cavity and those in the medium) are calculated using empirical approximations (see Reference 164 for review or 165 for recent developments). These terms are do not affect the SCF procedure since their dependence on electron density p is usually neglected. 

Ideally, one would like to smdy excited stales and ground states using wave functions of equivalent quality. Ground-state wave functions can very often be expressed in terms of a single Slater determinant formed from variationally optimized MOs, with possible accounting for electron correlation effects taken thereafter (or, in the case of DFT, the optimized orbitals that intrinsically include electron correlation effects are use in the energy functional). Such orbitals are determined in the SCF procedure. 

The process is continued for k cycles till we have a wavefunction and/or an energy calculated from that are essentially the same (according to some reasonable criterion) as the wavefunction and/or energy from the previous cycle. This happens when the functions i/ (l), i//(2),. .., j/(n) are changing so little from one cycle to the next that the smeared-out electrostatic field used for the electron-electron potential has (essentially) ceased to change. At this stage the field of cycle k is essentially the same as that of cycle k — 1, i.e. it is consistent with this previous field, and so the Hartree procedure is called the self-consistent-field-procedure, which is usually abbreviated as the SCF procedure. 

To expand a bit on Dewar s cautious endorsement of the SCF procedure [20] ( SCF calculations are by no means foolproof . ..Usually one finds a reasonably rapid convergence to the required solution ) occasionally a wavefunction is obtained that is not the best one available from the chosen basis set. This phenomenon is called wavefunction instability. To see how this could happen note that the SCF method is an optimization procedure somewhat analogous to geometry optimization (Section 2.4). In geometry optimization we seek a relative minimum or a transition state on a hypersurface in a mathematical energy versus nuclear coordinates space defined by E =/(nuclear coordinates) in wavefunction... 

Using the Roothaan-Hall Equations to do ab initio Calculations - the SCF Procedure... 

C(>I1J does not change because the SCF procedure refines the electron-electron repulsion (till the field each electron feels is consistent with the previous one), but H le in contrast represents only the contribution to the kinetic energy plus electron- nucleus attraction of the electron density associated with each pair of basis functions [Pg.212]

The zero subscripts in Eqs. (5.127) and (5.128) emphasize that the initial-guess c s, with no iterative refinement, were used to calculate G in the subsequent iterations of the SCF procedure Hcore will remain constant while G will be refined as the c s,... 

Step 8 - Comparing the density matrix from the latest c s with the previous density matrix to see if the SCF procedure has converged... 

As stated above, the following discussion applies to semiempirical methods that, like ab initio, use the SCF procedure and so pay some service to Eq. 6.1=5.82. 

---