Post Self-Consistent Field Calculations

For some systems a single determinant (SCFcalculation) is insufficient to describe the electronic wave function. For example, square cyclobutadiene and twisted ethylene require at least two configurations to describe their ground states. To allow several configurations to be used, a multi-electron configuration interaction technique has been implemented in HyperChem. 

HyperChem uses single determinant rather than spin-adapted wave functions to form a basis set for the wave functions in a configuration interaction expansion. That is, HyperChem expands a Cl wave function, in a linear combination of single Slater determ in ants 

The expansion coefficients Cj are determined by solving the Cl secular equation 

Evaluation of the Cl matrix elements is somewhat difficult. Fortunately, most matrix elements are zero because of the orthogonality of the MO s. There are only three types of non-zero elements which are needed to be computed. 

Configuration interaction (Cl) is a systematic procedure for going beyond the Hartree-Fock approximation. A different systematic approach for finding the correlation energy is perturbation theory 

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With the success of these calculations for isolated molecules, we began a systematic series of supermolecule calculations. As discussed previously, these are ab initio molecular orbital calculations over a cluster of nuclear centers representing two or more molecules. Self-consistent field calculations include all the electrostatic, penetration, exchange, and induction portions of the intermolecular interaction energy, but do not treat the dispersion effects which can be treated by the post Hartree-Fock techniques for electron correlation [91]. The major problems of basis set superposition errors (BSSE) [82] are primarily associated with the calculation of the energy. 

Nieuwpoort, W. C., D. Post, and P. Th. van Duijnen (1978). Calibration constant for Fe Mdssbauer isomer shifts derived from ab initio self-consistent-field calculations on octahedral FeFj, and Fe(CN)s clusters. Phys. Rev. B17, 91-98. 

P. J. Stephens, K. J. Jalkanen, F. J. Devlin, and C. F. Chabalowski,/. Phys. Chem., 97,6107 (1993). Ab Initio Calculation of Vibrational Circular Dichroism Spectra Using Accurate Post-Self-Consistent-Field Force Fields rivws-23-Dideuteriooxirane. 

Ab initio calculations can be performed at the Hartree-Fock level of approximation, equivalent to a self-consistent-field (SCF) calculation, or at a post Hartree-Fock level which includes the effects of correlation — defined to be everything that the Hartree-Fock level of approximation leaves out of a non-relativistic solution to the Schrodinger equation (within the clamped-nuclei Born-Oppenhe-imer approximation). 

How does a rigorously calculated electrostatic potential depend upon the computational level at which was obtained p(r) Most ab initio calculations of V(r) for reasonably sized molecules are based on self-consistent field (SCF) or near Hartree-Fock wavefunctions and therefore do not reflect electron correlation in the computation of p(r). It is true that the availability of supercomputers and high-powered work stations has made post-Hartree-Fock calculations of V(r) (which include electron correlation) a realistic possibility even for molecules with 5 to 10 first-row atoms however, there is reason to believe that such computational levels are usually not necessary and not warranted. The Mpller-Plesset theorem states that properties computed from Hartree-Fock wave functions using one-electron operators, as is T(r), are correct through first order (Mpller and Plesset 1934) any errors are no more than second-order effects. 

Density-Functional Theory. Transition metals pose a problem for classical quantum chemical methods like self-consistent field (SCF), perturbation theory, configuration interaction (Cl), and variations on these methods, because of the very large electron correlation. SCF underestimates binding substantially, and post-SCF methods are so expensive for transition metals that one can do a calculation only on models with few atoms. DFT on the other hand is relatively cheap it is about as expensive as SCF. Moreover, with the development of the generalized-gradient approximations it is also reasonably accurate. A large majority of quantum chemical... 

Quantum mechanics (QM) can be further divided into ab initio and semiempiri-cal methods. The ab initio approach uses the Schrodinger equation as the starting point with post-perturbation calculation to solve electron correlation. Various approximations are made that the wave function can be described by some functional form. The functions used most often are a linear combination of Slater-type orbitals (STO), exp (-ax), or Gaussian-type orbitals (GTO), exp (-ax2). In general, ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Self-consistency is achieved by a procedure in which a set of orbitals is assumed and the electron-electron repulsion is calculated. This energy is then used to calculate a new set of orbitals, and these in turn are used to calculate a new repulsion energy. The process is continued until convergence occurs and self-consistency is achieved. 

Quantum mechanical calculations are carried out using the Variational theorem and the Har-tree-Fock-Roothaan equations.t - Solution of the Hartree-Fock-Roothaan equations must be carried out in an iterative fashion. This procedure has been called self-consistent field (SCF) theory, because each electron is calculated as interacting with a general field of all the other electrons. This process underestimates the electron correlation. In nature, electronic motion is correlated such that electrons avoid one another. There are perturbation procedures whereby one may carry out post-Hartree-Fock calculations to take electron correlation effects into account. " It is generally agreed that electron correlation gives more accurate results, particularly in terms of energy. 

SM calculations are broadly based on either the (i) Hartree-Fock method (ii) Post-Hartree-Fock methods like the Mpller-Plesset level of theory (MP), configuration interaction (Cl), complete active space self-consistent field (CASSCF), coupled cluster singles and doubles (CCSD) or (iii) methods based on DFT [24-27]. Since the inclusion of electron correlation is vital to obtain an accurate description of nearly all the calculated properties, it is desirable that SM calculations are carried out at either the second-order Mpller-Plesset (MP2) or the coupled cluster with single, double, and perturbative triple substitutions (CCSD(T)) levels using basis sets composed of both diffuse and polarization functions. 

Table 1 contains some further information useful to characterize the different contributions to the molecule/surface interaction orientation dependence and the typical strength of the different contributions, and whether or not they can be understood on a purely classical basis. If one wants to calculate molecule/surface interactions by means of quantum-mechanical or quantum-chemical methods, the most important question is whether standard density functional (DPT) or Hartree-Fock theory (self consistent field, SCF) is sufficient for a correct and reliable description. Table 1 shows that all contributions except the Van der Waals interaction can be obtained both by DPT and SCF methods. However, the results might be connected with rather large errors. One famous example is that the dipole moment of the CO molecule has the wrong sign in the SCF approximation, with the consequence that SCF might yield a wrong orientation of CO on an oxide surface (see also below). In such cases, the use of post Hartree-Fock methods or improved functionals is compulsory. 

Results from self-consistent field (SCF) molecular orbital calculations, in combination with gas-phase photoelectron data and results from post-SCF calculations have provided a basis for descriptions of the valence electronic structure of gas-phase nucleotides and of nucleotides in water-counterion clusters. These descriptions contain values for 11 to 14 of the lowest energy ionization events in the DNA nucleotides 5 -dGMP , 5 -dAMP, 5 -dCMP and 5 -dTMP . When used with an evaluation of the difference between the Gibbs free energies of hydration for the initial and final states associated with ionization, this approach also describes the influence of hydration on the energetic ordering of ionization events in nucleotides. 

In this section we will introduce some wavefunction-based methods to calculate photoabsorption spectra. The Hartree-Fock method itself is a wavefunction-based approach to solve the static Schrodinger equation. For excited states one has to account for time-dependent phenomena as in the density-based approaches. Therefore, we will start with a short review of time-dependent Hartree-Fock. Several more advanced methods are available as well, e.g. configuration interaction (Cl), multireference configuration interaction (MRCI), multireference Moller-Plesset (MRMP), or complete active space self-consistent field (CASSCF), to name only a few. Also flavours of the coupled-cluster approach (equations-of-motion CC and linear-response CQ are used to calculate excited states. However, all these methods are applicable only to fairly small molecules due to their high computational costs. These approaches are therefore discussed only in a more phenomenological way here, and many post-Hartree-Fock methods are explicitly not included. 

The functional was defined only for closed-shell systems. In fact, original vdW-DFl computation was just a post-SCF (self-consistent field) procedure, where the revPBE SCF procedure was done first, then the revPBE correlation part except for LDa subtracted from the total energy and J was calculated from the revPBE... 

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