Energy ordering

For qualitative insight based on perturbation theory, the two lowest order energy eorreetions and the first-order wavefunetion eorreetions are undoubtedly the most usetlil. The first-order energy eorresponds to averaging the eflfeets of the perturbation over the approximate wavefunetion Xq, and ean usually be evaluated without diflfieulty. The sum of aJ, Wd ds preeisely equal to tlie expeetation value of the Hamiltonian over... 

In all methods, the first-order interaetion energy is just the differenee between the expeetation value of the system Hamiltonian for the antisyimnetrized produet fiinetion and the zeroth-order energy... 

Figure Al.5.3 shows that, as in interactions between other species, the first-order energy for Fte-Fle decays exponentially with interatomic distance. It can be fitted [70] within 0.6% by a fimction of the fonn... 

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. 

While all contributions to the spin Hamiltonian so far involve the electron spin and cause first-order energy shifts or splittings in the FPR spectmm, there are also tenns that involve only nuclear spms. Aside from their importance for the calculation of FNDOR spectra, these tenns may influence the FPR spectnim significantly in situations where the high-field approximation breaks down and second-order effects become important. The first of these interactions is the coupling of the nuclear spin to the external magnetic field, called the... 

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. 

The first-order energy correction with respect to the unperturbed problem is then... 

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 zeroth-order vibronic wave function is ut- Uy 0 0-I-). The zeroth-order energy is... 

The zeroth-order energy level is twofold degenerate. The corresponding vibronic basis functions are ur K+2 0 0 —) = 11) and luj- A"—2 0 0 +) = 2). The first-order energy correction is... 

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

In other cases, the zeroth-order vibronic levels are generally more than twofold degenerate and the perturbative handling is much more complicated. An exception is the case 07= , U( =li K = 0 with the twofold degenerate zeroth-order level. The basis functions are 1 1 1 1 —) = 1) and 1 —1 1 —l- -)s 2). The zeroth-order energy is... 

Hartree-Fock wavefunction, is an eigenfunction of and the corresponding oth-order energy Eg° is equal to the sum or orbital energies for the occupied molecular... 

The sum of the zeroth-order and first-order energies thus corresponds to the Hartree-Fock energy (compare with Equation (2.110), which gives the equivalent result for a closed-shell system) ... 

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 first-order energy eorreetion is given in terms of the zeroth-order (i.e., unperturbed) wavefunetion as ... 

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 ... 

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