First-order perturbation energy

If the perturbation function shows cubic symmetry, and in certain other special cases, the first-order perturbation energy is not effective in destroying the orbital magnetic moment, for the eigenfunction px = = i py leads to the same first-order perturbation terms as pi or pv or any other combinations of them. In such cases the higher order perturbation energies are to be compared with the multiplet separation in the above criterion. 

The first order perturbation energy is calculated by the following zeroth-order approximation to the ground-state wave function P° of the two molecules ... 

Before we tarn to MO theory of molecular interactions a short discussion on the reliability of semiempirical calculations of the CNDO type by means of perturbation theory would be useful. For a better understanding of the possibilities and limitations of semiempirical MO approaches to intermolecular forces we calculated first-order perturbation energies for very simple complexes with and... 

Hence, the first-order perturbed energy can be written as... 

In the excimer configuration the excited molecular states lZ, and 1Lb are each split into nondegenerate exciton states as shown schematically in Figure 11. The first-order perturbation energy A,Eexc at the equilibrium interplanar separation R0 is given by66... 

It turns out that the most significant contribution to the total nuclear spin—spin interaction in most cases is the contact term H3. Since the first-order perturbation energy due to H3 is zero,9 the energy of interaction between nuclei A and X, namely AX, is represented by the second-order perturbation energy... 

The type of correlated method that has enjoyed the most widespread application to H-bonded systems is many-body perturbation theory, also commonly referred to as Mpller-Plesset (MP) perturbation theory This approach considers the true Hamiltonian as a sum of its Hartree-Fock part plus an operator corresponding to electron correlation. In other words, the unperturbed Hamiltonian consists of the interaction of the electrons with the nuclei, plus their kinetic energy, to which is added the Hartree-Fock potential the interaction of each electron with the time-averaged field generated by the others. The perturbation thus becomes the difference between the correct interelectronic repulsion operator, with its instantaneous correlation between electrons, and the latter Hartree-Fock potential. In this formalism, the Hartree-Fock energy is equed to the sum of the zeroth and first-order perturbation energy corrections. 

In this case, the first-order perturbation energy upon dimer formation would simply be the Coulomb energy Ec between the unperturbed charge distributions on the monomers and M2. ... 

This expression for the perturbation energy can be very simply described The first-order perturbation energy for a non-degenerate state of a system is just the perturbation function averaged over the corresponding unperturbed state of the system. 

The first-order perturbation energy W is the average value of the perturbation function H = e2/rn over the unperturbed state of the system, with the value... 

The methods which we have used in Section 23 to obtain the first-order perturbation energy are not applicable when the energy level of the unperturbed system is degenerate, for the reason that in carrying out the treatment we assumed that the perturbed wave function differs only slightly from one function which is the solution of the unperturbed wave equation for a given energy value whereas now there are several such functions, all... 

The first-order perturbation energy is zero, as can be seen from inspection of the perturbation function. 

In general, the physical properties of an electron system are defined by referring to a specific perturbation problem and can be classified according to the order of the perturbation effect. For instance, the electric dipole moment is associated with the first-order response to an applied electric field (i.e. the perturbation), the electric polarizability with the second-order response, hyperpolarizabilities with higher-order terms. In addition to dipole moments, there is a number of properties which can be calculated as a first-order perturbation energy and identified with the expectation value... 

Figure 4. The difference between the Dirac energy and the lORA energy plus the first-order perturbation energy correction as a function of nuclear charge. The same graph is also shown for the ERA+PT calculations.

The accuracy of the quasi-relativistic energies and the first-order perturbation energy corrections have been studied by performing ERA and ERA+PT calculations on the lowest. Vj 2 of using a few values for 7. The results of the ERA and ERA+PT calculations are summarized in Figure 5 and Figure 6, respectively. 

The first-order perturbation energy correction to the ERA energies (in Hartrees) of the 4 lowest s states of calculated using the scaling factors y of 1.0 and 1.24, respectively. The total ERA-t-PT energies are compared to the corresponding Dirac energies. 

Figure 7. The optimal scaling factor y of the l-Sjyj one-electron state obtained using the ERA model and using the ERA model corrected for the first-order perturbation energy (ERA+PT), respectively, as a function of the nuclear charge.

However, since the perturbing Hamiltonian consists of only electron-repulsion terms, we find the first-order perturbation energy exactly corrects for the omission ... 

Appendix E. Analysis of the first-order perturbation energy References... 

APPENDIX E. ANALYSIS OF THE FIRST-ORDER PERTURBATION ENERGY... 

Eo is the energy at Qo, the next two terms are the first-order perturbation energy, and the last term is the second-order perturbation energy. While Eq. (3) is valid only for Q very small, we can select Qo anywhere on Fig. 1. Hence Eq. (3) is general for the purpose of displaying symmetry properties. 

Recalling that equals 13.604 eV when the He reduced mass is used and putting Z = 2, we find for the first-order perturbation energy correction for the helium ground state ... 

Since we are assuming unequal roots, the quantities H j2 H[i,..., H 44 - are all nonzero.Therefore,C2 = 0, C3 = 0,..., q = O.ITie normalization condition (9.88) gjves Cl = l.The correct zeroth-order wave function corresponding to the first-order perturbation energy correction /fj, is then [Eq. (9.76)] For the root H 22, the same... 

For helium the first-order perturbation energy correction is e /ryi averaged over... 

These can be considered as a perturbation, the unperturbed problem being the harmonic oscillator. Since the average value of QkQiQm is zero in any state v, the first-order perturbation energy due to the cubic terms vanishes, but the second-order energy does not. The first-order energy from the quartic terms involves the mean value of hkbnnQkQiQ,aQn, which vanishes except for two classes of terms QIQ and QjJ. The mean values of terms of the first class arc given by (see Appendix III)... 

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