Perturbation theoiy

Another distinguishing aspect of MO methods is the extent to which they deal with electron correlation. The Hartree-Fock approximation does not deal with correlation between individual electrons, and the results are expected to be in error because of this, giving energies above the exact energy. MO methods that include electron correlation have been developed. The calculations are usually done using MoUer-Plesset perturbation theoiy and are designated MP calculations." ... 

The present paper is aimed at developing an efficient CHF procedure [6-11] for the entire set of electric polarizabilities and hyperpolarizabilities defined in eqs. (l)-(6) up to the 5-th rank. Owing to the 2n+ theorem of perturbation theoiy [36], only 2-nd order perturbed wavefunctions and density matrices need to be calculated. Explicit expressions for the perturbed energy up to the 4-th order are given in Sec. IV. 

Carbon tetrachloride solution, 409, 446, 460 Carbonic anhydrase, 481 CASPT2, see Perturbation theoiy,... 

When E > Et, the only second-order perturbation theoiy contribution comes from the first term of Eq. (3.39), Thus,... 

A second theoretical approach to understanding E- V transfer is based on the notion that long-range attractive forces might couple the electronic and vibrational degrees of freedom. Ewing has developed a theory for such coupling that is similar to the theory for V- V transfer proposed earlier by Sharma and Brau. First order perturbation theoiy is used in this approach to calculate the probability for the E->V transition ... 

It has been shown, that the first two terms in the last formula also follow from perturbation theoiy. However, in this perturbation theory the unperturbed operator does not correspond to the free molecule, but in addition contains a harmonic oscillator potential (with the angle 8 as the eorresponding coordinate). The anharmonidly is treated as a perturbation. 

Iterative scheme of the symmetry adapted perturbation theoiy... 

Let us recall that the electrostatic interaction energy Ee]st(.ABC) of the ground-state ha =0,riB = 0, nc = 0) molecules is defined as the first-order correction to the energy in the polarization approximation perturbation theoiy ... 

The E (ABC) term in qfmmetiy-adapted perturbation theoiy represents onty pan of the fiist-order correction to the energy (the rest being the valence repulsion). 

The expression for the three-bo(ty non-additivity of the valence repulsion [given by formula (f3.39), based on definition (13.37) of the first-order correction in symmetry-adapted perturbation theoiy and from definition (13.52) of the three-body contribution] is ... 

Dyall, K.G. Second-order Moller-Plesset perturbation theoiy for molecular Dirac-Hartree-Foch wavefunctions. Theory for up to two open-shell electrons. Chem. Phys. Lett. 224, 186-194 (1994)... 

In general terms, since an exact solution of the vibrational equation in terms of anharmonic wave function is not possible, use is made of the fact at for finite displacements at each step of the potential energy or transitional dipole moment expansion, the higher terms are much smaller than the respective lower terms. A perturbation theoiy treatment becomes, therefore, feasible. The potential energy may be expressed in the form [3]... 

The independent Cartesian components of the polarizability (a ) and hyperpolarizability (ya, ) are given in Table 1. The method-dependence of the mean and the anisotropy of the dipole polarizability and the mean second hyperpolarizability are shown in Figs. 1, 2, and 3, respectively. Figiu e 1 shows clearly that electron correlation reduces the size of the mean dipole polarizability. The sequence a(SCF) < a(MP2) < a(MP3) < a(MP4) shows a monotonic decrease of the values calculated with Mpller-Plesset perturbation theoiy. Our best theoretical value a(CCSD(T)) is quite close to a(MP2) and a(MP3). For comparison, we show on the same figure the DFT values for the mean calculated with mPWlPW91 and B3PW91. The former is close to the MP4 value and the latter close to SDQ-MP4. Figure 2 shows that the method-dependence of the dipole polarizability... 

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