SCF orbitals

In the MPPT/MBPT method, once the reference CSF is chosen and the SCF orbitals belonging to this CSF are detennined, the wavefiinction T and energy E are detennined in an order-by-order maimer. The perturbation equations determine what CSFs to include and their particular order. This is one of the primary strengdis of this technique it does not require one to make fiirtlier choices, in contrast to the MCSCF and Cl treatments where one needs to choose which CSFs to include. 

The basic self-consistent field (SCF) procedure, i.e., repeated diagonalization of the Fock matrix [26], can be viewed, if sufficiently converged, as local optimization with a fixed, approximate Hessian, i.e., as simple relaxation. To show this, let us consider the closed-shell case and restrict ourselves to real orbitals. The SCF orbital coefficients are not the... 

A CASSCF calculation is a combination of an SCF computation with a full Configuration Interaction calculation involving a subset of the orbitals. The orbitals involved in the Cl are known as the active space. In this way, the CASSCF method optimizes the orbitals appropriately for the excited state. In contrast, the Cl-Singles method uses SCF orbitals for the excited state. Since Hartree-Fock orbitals are biased toward the ground state, a CASSCF description of the excited state electronic configuration is often an improvement. 

The Hartree-Fock equations form a set of pseudo-eigenvalue equations, as the Fock operator depends on all the occupied MOs (via the Coulomb and Exchange operators, eqs. (3.36) and (3.33)). A specific Fock orbital can only be determined if all the other occupied orbitals are known, and iterative methods must therefore be employed for determining the orbitals. A set of functions which is a solution to eq. (3.41) are called Self-Consistent Field (SCF) orbitals. 

Figure 9. Determination of the first electron affinity, and the first and higher ionization potentials of formyl radical from the SCF orbital energies and electronic repulsion integrals, and K,j (cf. eqs. (90), (92), and (93)). The experimental value (112), 9.88 eV, for the first ionization potential corresponds to the theoretical value I . All entries are given in eV. With A and I a lower index stands for MO the upper one indicates the state multiplicity after ionization.

Let us note that the two conditions ej,=0 and < Pj >=0 (i 7 j) can be satisfied only with canonical SCF orbitals. Thus, in fact, the present theory can be applied only in such cases. However it has been demonstrated (12) that in most systems, the strongly occupied MCSCF orbitals and the SCF orbitals are extremely close one to the others. Therefore, in practice, the present theory also applies to the strongly occupied MCSCF orbitals. 

We have demonstrated formally that the optimum orbitals of any given molecular system (canonical SCF orbitals or strongly occupied MCSCF orbitals that are closed to the SCF ones) can be described very simply in the regions surrounding each nucleus... 

These results show that convergence is favored when the basis used are either the l-SRH ox the Absar and Coleman [32,33] Reduced Hamiltonian (RH) eigen-orbitals. The other two basis used were the SCF orbitals and the core Hamiltonian eigen-orbitals Core H)... 

Daudey J-P (1974) Direct determination of localized SCF orbitals. Chem Phys Lett 24 574... 

Finally, it should be stressed that although the spectroscopic data yield the order of one-electron core energies as 5 Ko ir, these quantities are not synonymous with the corresponding SCF orbital energies. Consequently there is no contradiction involved by the... 

For the moment however it is not clear where the results of X - a calculations can be fitted into the pattern since it is not evident that the calculated electronic energies of the various levels have the same significance viz a viz Koopmans theorem as do the SCF orbital energies. 

Following Hendrickson s (55) treatment of Fe(Cp)2 one may write the SCF orbital energies as... 

For the open-shell d3 V(Cp)2 system, with the 4Z (a S2) ground level, the SCF orbital energies may be similarly expressed without ambiguity, yielding AeSCF =... 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

In this case, it can be proved that the canonical SCF orbitals, being solutions of Eq. (26), are symmetry orbitals, i.e. that they belong to irreducible representations of the symmetry group. 12) If the number of molecular orbitals is larger than the dimension of the largest irreducible representation of the symmetry group, it must then be concluded that the set of all N molecular orbitals form a reducible representation of the group which is the direct sum of all the irreducible representations spanned by the CMO s. 

Koopmann s theorem establishes a connection between the molecular orbitals of the 2jV-electron system, just discussed, and the corresponding (2N- 1 Electron system obtained by ionization. The theorem states If one expands the (2N - 1) molecular spin-orbitals of the ground state of the ionized system in terms of the 2N molecular spin-orbitals of the ground state of the neutral system, then one finds that the orbital space of the ionized system is spanned by the (2N - 1) canonical orbitals with the lowest orbital energies ek i.e. to this approximation the canonical self-consistent-field orbital with highest orbital energy is vacated upon ionization. This theorem holds only for the canonical SCF orbitals. 13>... 

The PNO extrapolations in Fig. 4.8 and Table 4.6 require localization of the occupied SCF orbitals to ensure size-consistency. In order to preserve this size-consistency for the CBS PNO extrapolations, we have restricted these (Zmax + f)-3 extrapolations to a linear form, Eq. (6.2). The new double extrapolation employs this linear extrapolation of pairs of CBS2/cc-pVnZ calculations and thus is rigorously size-consistent. Note that the nonlinear N-parameter (Zmax + a)-" extrapolations using least-squares fits to more than N cc-pVnZ energies are not size-consistent [53,55],... 

If we choose only one determinant built from the lowest /2 SCF-orbitals, the "configuration interaction method will naturally give us Wq = Ao with the energy eigenvalue q as the best groimd-state description. This is clearly identical with the SCF result of the last section. 

The wave function /lo constructed from SCF orbitals is "so good that it cannot be improved by the inclusion of singly excited determinants. The main effect of increasing the number of singly excited determinants in the Cl problem will be a better description of the excited state levels. 

The number No of occupied valence SCF orbitals in a molecule is typically less than the total number Nmb of orbitals in the minimal valence basis sets of all atoms. The full valence MCSCF wavefunction is the optimal expansion in terms of all configurations that can be generated from N b molecular orbitals. Closely related is the full MCSCF wavefunction of all configurations that can be generated from Ne orbitals, where Nc is the number of valence electrons, i.e. each occupied valence orbital has a correlating orbital, as first postulated by Boys (48) and also presumed in perfect pairing models (49,50), We shall call these two types of frill spaces FORS 1 and FORS 2. In both, the inner shell remains closed. 

The present approach will be generalized in two respects. On the one hand, the a priori truncation method will be extended to quintuply and sextuply excited configurations. On the other hand, the truncation method as well as the dynamic correlation estimate will be extended to systems where the number of strongly occupied orbitals exceeds the number of SCF orbitals, entailing zeroth-order wavefunctions that are dominated by several determinants. It will then be possible to combine the two approaches. 

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... 

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