Perturbation theory expression for

It has been well known for some time (e.g. [36]) that the next component in importance is that of connected triple excitations. By far the most cost-effective way of estimating them has been the quasiper-turbative approach known as CCSD(T) introduced by Raghavachari et al. [37], in which the fourth-order and fifth-order perturbation theory expressions for the most important terms are used with the converged CCSD amplitudes for the first-order wavefunction. This account for substantial fractions of the higher-order contributions a very recent detailed analysis by Cremer and He [38] suggests that 87, 80, and 72 %, respectively, of the sixth-, seventh-, and eighth-order terms appearing in the much more expensive CCSDT-la method are included implicitly in CCSD(T). 

The well-known perturbation theory expression for the non-adiabatic rate constant is given by (25, 42-45)... 

Perturbation theory expressions for Wc (r), Wt (r), Uc[p] and Tc[p] can be derived but we do not give these here. We° note, however, that the KS correlation potential vc(r) commences19 in second order in the coupling constant. The term quadratic in the coupling constant vc2(r) is then... 

In two previous papers [8,9] we have calculated the static polarizabilities and hyperpolarizabilities for ls3p Pj (J = 0, 2)-states of helium. The method was based on degenerate perturbation-theory expressions for these quantities. The necessary dipole matrix elements were found by using the high-precision wave function on framework of the configuration-interaction (Cl) method [10]. The perturbed wave functions are also expanded in a basis of accurate variational eigenstates [11]. These basis sets of the wave functions explicitly take account of electron correlation. To control the result we have also carried out similar calculations with Fues model potential method. 

The fourth-order time-dependent perturbation theory expression for P(3)(co0) is... 

The time-dependent perturbation theory expression for the first-order amplitude on the excited-state surface, n(t), is given by... 

Comparing calculated and experimentally determined reactivity worth enables verification of the accuracy qf nuclear data and the adequacy of computational methods used. For a meaningful comparison of theory and experiment, it is essential that the perturbation theory expressions used for the calculations apply to exactly the same parameter as that deduced fi-om the experiments, and that these expressions are evaluated accurately. This paper reviews three aspects of accurate determination of reactivity (1) the definition of reactivity, (2) high-order perturbation theory expressions for reactivity, and (3) the accuracy of computational techniques based on the multigroup approximation. 

Exact perturbation theory expressions for the static reactivity can be obtained from the pair of equations (78)-(81)and (79)-(80) [see derivation ofEq. (8)],... 

Different perturbation theory expressions for reactivity are obtained from different formulations of the neutron transport equation. 

The exact perturbation theory expression for the static reactivity, Eq. (11), can be expressed in terms of the unperturbed flux and the flux perturbation as follows ... 

An exact perturbation theory expression for the reactivity in an altered system is... 

Generalized-function formulations of GPT for homogeneous systems are the source of sensitivity functions for different integral parameters Equation (189) for reactivity worths, and Eq. (162) for ratios of linear and bilinear functionals. The first-order perturbation theory expression for reactivity [Eq. (132)] can also be used for sensitivity studies. 

One possible way to include mode-mode couplings in normal coordinates is by perturbation theory. The perturbation-theory expressions for the energies of a polyatomic system are usually given in terms of dimensionless normal coordinates, m=l, 2, . . . , F-1. These are-... 

In the sense of the second-order perturbation theory expression for the energy, only those orbitals have to be considered, because for the group VIII metals in the last columns of the periodic system they are the only ones to have a finite density of states at Ep. 

Let us start with the first definition as derivatives of the energy, Eqs. (4.65) to (4.67). Again we will use the perturbation theory expression for the perturbed energy, Eq. (3.15), but differentiate it now twice with respect to the appropriate components of the field or field gradient. This leads us immediately to the second-order correction to the energy, because the first-order correction depends only linearly on the fields. We can therefore express the polarizabilities as... 

The second-order perturbation theory expression for a component of the spin rotation tensor then becomes... 

Table C.l Operators and prefactors for the exact first-order Rayleigh-Schrodinger perturbation theory expressions for molecular properties. See Ek[. (C.l).

If the energy and angular momentum of the diatomic are not too large and the potential energy function for the diatomic is either the harmonic or Morse function, n can be determined from the following second-order perturbation theory expression for Ed( ,7) ... 

Independently, the vibrational lifetime can be estimated by using classical equilibrium MD to approximate the quantum first-order perturbation theory expression for the vibrational relaxation rate (inverse of the lifetime). The quantum relaxation rate of a vibrational mode coupled to a bath is proportional to the Fourier transform of a force along the vibrational coordinate correlation function. The idea is to approximate this correlation function using classical equilibrium MD trajectories. " " ... 

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