Energy second-order

The details of the second-order energy depend on the fonn of exchange perturbation tiieory used. Most known results are numerical. However, there are some connnon features that can be described qualitatively. The short-range mduction and dispersion energies appear in a non-expanded fonn and the differences between these and their multipole expansion counterparts are called penetration tenns. 

In other words, for calculating the second-order energy (the vibrational energy), we only have to keep the term to do with the interatomic distance. The other terms, then, will enter the total Schrddinger equation in higher orders. 

Zeroth-order level, corresponding to the vibronic state l)r (r Uc (c +) = Ur Ur uc Uc +) is nondegenerate. The zeroth, first- and second-order energies are... 

The second-order energy corrections have the form (B.8) with... 

The il/j in Equation (3.21) will include single, double, etc. excitations obtained by promoting electrons into the virtual orbitals obtained from a Hartree-Fock calculation. The second-order energy is given by ... 

The second-order energy correction is expressed as follows ... 

When this result is used in the earlier expression for the second-order energy correction, one obtains ... 

The second-order energy correction can be evaluated in like fashion by noting that... 

Examination of the energy expression 17.7 shows that the polarizability components are the second-order energies and a little analysis shows that (for example)... 

The last equation shows that the second-order energy correction may be written in terms of the first order wave function (c,) and matrix elements over unperturbed states. The second-order wave function correction is... 

Let us look at the expression for the second-order energy correction, eq. (4.38). This involves matrix elements of the perturbation operator between the HF reference and all possible excited states. Since the perturbation is a two-electron operator, all matrix elements involving triple, quadruple etc. excitations are zero. When canonical HF... 

Ho is the normal electronic Hamilton operator, and the perturbations are described by the operators Pi and P2, with A determining the strength. Based on an expansion in exact wave functions, Rayleigh-Schrddinger perturbation theory (section 4.8) gives the first- and second-order energy collections. 

It will be seen that the second-order treatment leads to results which deviate more from the correct values than do those given by the first-order treatment alone. This is due in part to the fact that the second-order energy was derived without considerar-tion of the resonance phenomenon, and is probably in error for that reason. The third-order energy is also no doubt appreciable. It can be concluded from table 3 that the first-order perturbation calculation in problems of this type will usually lead to rather good results, and that in general the second-order term need not be evaluated. 

We here see the structure of the second-order energy. All the Ho, 6, and 7 0 are connected to each other, but A is not necessarily connected to each of those. As pointed out, the term involving in Eqn. (15) vanishes, as long as the A equation is solved in the same excitation manifold as the CC equation. However, the term involving in Eqn. (15) creates a... 

Moreover, if the wave function + Xxp P is used as a trial function 0, then the quantity W from equation (9.2) is equal to the second-order energy determined by perturbation theory. Any trial function 0 with parameters which reduces to -h 20o for some set of parameter values yields an approximate energy W from equation (9.2) which is no less accurate than the second-order perturbation value. 

The only nonvanishing matrix elements of x3 are those with j = n 1 and j = n 3. This result is obtained by repeated application of Eq. (40), as before. Thus, there are four terms that the cubic potential constant contributes to the second-order energy correction, Eq, (35). The final result can be written as... 

As electric fields and potential of molecules can be generated upon distributed p, the second order energies schemes of the SIBFA approach can be directly fueled by the density fitted coefficients. To conclude, an important asset of the GEM approach is the possibility of generating a general framework to perform Periodic Boundary Conditions (PBC) simulations. Indeed, such process can be used for second generation APMM such as SIBFA since PBC methodology has been shown to be a key issue in polarizable molecular dynamics with the efficient PBC implementation [60] of the multipole based AMOEBA force field [61]. 

Thus the zero-, first-, and second-order energies are ... 

The formal treatment is quite similar to the derivation of the principal g values as developed in Eqs. (7C) through (18C). The second-order energy term is set equal to the hyperfine term from the spin Hamiltonian, and for the z direction... 

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