Numerical calculations perturbation theory

The second class of theories can be characterized as attempts to find approximate solutions to the Schrodinger equation of the molecular complex as a whole. Two approaches became important in numerical calculations perturbation theory (PT) and molecular orbital (MO) methods. 

There are several different ways in which quantum mechanics has been applied to the problem of relating the barrier to the frequency separation of the spectroscopic doublets. These are all approximation procedures and each is especially suitable under an appropriate set of circumstances. For example one may use perturbation theory, treating either the coupling of internal and external angular momenta, the molecular asymmetry, or the potential barrier as perturbations. Some of the different treatments have regions of overlap in which they give equivalent results choice is then usually made on the basis of convenience or familiarity. Extensive numerical tabless have been prepared which simplify considerably the calculations. 

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 various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

Feg). Subsequently, thermodynamic properties of spins weakly coupled by the dipolar interaction are calculated. Dipolar interaction is, due to its long range and reduced symmetry, difficult to treat analytically most previous work on dipolar interaction is therefore numerical [10-13]. Here thermodynamic perturbation theory will be used to treat weak dipolar interaction analytically. Finally, the dynamical properties of magnetic nanoparticles are reviewed with focus on how relaxation time and superparamegnetic blocking are affected by weak dipolar interaction. For notational simplicity, it will be assumed throughout this section that the parameters characterizing different nanoparticles are identical (e.g., volume and anisotropy). 

The instantaneous OH frequency was calculated at each time step in an adiabatic approximation (fast quantal vibration in a slow classical bath ). We applied second-order perturbation theory, in which the instantaneous solvent-induced frequency shift from the gas-phase value is obtained from the solute-solvent forces and their derivatives acting on a rigid OH bond. This method is both numerically advantageous and allows exploration of sources of various solvent contributions to the frequency shift. 

Working out the parabolic wavefunctions in terms of the Laguerre polynomials is useful in the analytic calculation of the Stark effect using perturbation theory. However, it is not useful in very strong electric fields. Here we outline a more general procedure which is valid in strong fields and lends itself to numerical computations. If we replace jq( ) and u2(rj) in Eqs. (6.8a) and (6.8b) by1-3 15... 

Most numerical methods for calculating molecular hyperpolarizability use sum over states expressions in either a time-dependent (explicitly including field dependent dispersion terms) or time-independent perturbation theory framework [13,14]. Sum over states methods require an ability to determine the excited states of the system reliably. This can become computationally demanding, especially for high order hyperpolarizabilities [15]. An alternative strategy adds a finite electric field term to the hamiltonian and computes the hyperpolarizability from the derivatives of the field dependent molecular dipole moment. Finite-field calculations use the ground state wave function only and include the influence of the field in a self-consistent manner [16]. 

For the HgH system numerical wavefunctions were obtained for Hg using both relativistic (Desclaux programme87 was used) and non-relativistic hamiltonians. The orbitals were separated into three groups an inner core (Is up to 3d), an outer core (4s—4/), and the valence orbitals (5s—6s, 6p). The latter two sets were then fitted by Slater-type basis functions. This definition of two core regions enabled them to hold the inner set constant ( frozen core ) whilst making corrections to the outer set, at the end of the calculation, to allow some degree of core polarizability. The correction to the outer core was done approximately via first-order perturbation theory, and the authors concluded that in this case core distortion effects were negligible. 

We found a correct account of the broadening magnitude and increase with temperature, out of reach of the renormalized perturbation theory. Exact numerical calculations by Schreiber and Toyozawa53 agree with our data and confirm our values of the exciton-phonon coupling strengths. 

The energies of the levels in an electric field can be calculated by numerical diagonalisation of the above matrix for different values of the electric field and the J, M quantum numbers. However, perturbation theory has also often been used and we may readily derive an expression for the second-order Stark energy using the above matrix elements. The result is as follows ... 

The pioneering work on the application of the many-body perturbation theory to atomic and molecular systems was performed by Kelly.5-17-21 He applied the method to atoms using numerical solutions of the Hartree-Fock equations. Many other calculations on atomic systems were subsequently... 

---