Perturbation theory coefficients

Table 7 Perturbation theory coefficients E for expansion of the energies in powers of R for the Is, 2p, 3d, 4f and 5g levels of a hydrogen-like atom confined in a spherical box of radius R (see Equation (51) of text). The numbers in parentheses indicate the power of 10... 

Perturbation theory yields a siim-over-states fomnila for each of the dispersion coefficients. For example, the isotropic coefficient for the interaction between molecules A and B is given by... 

The importance of FMO theory hes in the fact that good results may be obtained even if the frontier molecular orbitals are calculated by rather simple, approximate quantum mechanical methods such as perturbation theory. Even simple additivity schemes have been developed for estimating the energies and the orbital coefficients of frontier molecular orbitals [6]. 

These phenomenological level-to-level rate coefficients are related to the state-to-state Ri f coefficients derived by applying perturbation theory to the electromagnetic perturbation through... 

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

If we used perturbation theory to estimate the expansion coefficients c etc., then all the singly excited coefficients would be zero by Brillouin s theorem. This led authors to make statements that HF calculations of primary properties are correct to second order of perturbation theory , because substitution of the perturbed wavefunction into... 

To prove this let us make more precise the short-time behaviour of the orientational relaxation, estimating it in the next order of tfg. The estimate of U given in (2.65b) involves terms of first and second order in Jtfg but the accuracy of the latter was not guaranteed by the simplest perturbation theory. The exact value of I4 presented in Eq. (2.66) involves numerical coefficient which is correct only in the next level of approximation. The latter keeps in Eq. (2.86) the terms quadratic to emerging from the expansion of M(Jf ). Taking into account this correction calculated in Appendix 2, one may readily reproduce the exact... 

The coefficient of the 8-function reflects the pile-up of the two-level systems that would have had a value of e < S were it not for quantum effects. These fast two-level systems will contribute to the short-time value of the heat capacity in glasses. The precise distribution in Eq. (69) was only derived within perturbation theory and so is expected to provide only a crude description of the interplay of clasical and quantum effects in forming low-barrier TLS. Quantitative discrepancies from the simple perturbative distribution may be expected owing to the finite size of a tunneling mosaic cell, as mentioned earlier. 

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

Next, fin is introduced and viewed as a weak perturbation. Given the just described nodal properties of HOMO and LUMO, all of the / 14 integrals in the chain will act in the same direction, which is then easy to predict using first-order perturbation theory. In the HOMO, any two coefficients that are in a 1-4 relation... 

In other words, the diagonal elements of the perturbing Hamiltonian provide the first-order correction to the energies of the spin manifold, and the nondiagonal elements give the second-order corrections. Perturbation theory also provides expressions for the calculation of the coefficients of the second-order corrected wavefunctions l / in terms of the original wavefunctions (p)... 

To invoke the perturbation theory for a small anharmonic coupling coefficient, we use the Wick theorem for the coupling of the creation and annihilation operators of low-frequency modes in expression (A3.19). Retaining the terms of the orders y and y2, we are led to the following expressions for the shift AQ and the width 2T of the high-frequency vibration spectral line 184... 

The term angular overlap model was first used in 1965 to describe a more general treatment, which may also be called the model (12,13). The relative energies of the orbitals in a partly-filled /-shell are expressed in terms of the parameters ex (X = a, rr, etc.), whose coefficients can be calculated from the geometry of the system. The eK model was developed from perturbation theory, but is equivalent to the E2 model when only a-overlap is considered, and to the Yamatera-McClure model for orthoaxial chromophores with linear ligators. The notation ex is often used, where ... 

Perturbation theory like MP2 or CASPT2 should be used only when the perturbation is small. Orbitals that give rise to large coefficients for the... 

Dunham [5] derived these expressions Y ((t)e,Be,a0, necessarily manually, through a JBKW procedure, which he claimed to make more general [4] than what had appeared in previous literature. Dunham reported expressions F containing coefficients aj up to a, and Sandeman [19] and Woolley [20] extended manually these results according to a roughly analogous procedure. Kilpatrick [21] applied perturbation theory in successive orders to derive expressions for 1, and Bouanich [22] applied Rayleigh-Ritz perturbation theory for solution of... 

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