Function wave, perturbed

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

The parameter is introduced to keep track of the order of the perturbation series, as will become clear. Indeed, one can perform a Taylor series expansion of the perturbed wave functions and perturbed energies using X to keep track of the order of the expansions. Since the set of eigenfunctions of the unperturbed SE form a complete and orthonormal set, the perturbed wave functions can be expanded in terms of them. Thus,... 

The parameter X has been embedded in the definition of Hp. The wave function from perturbation theory [equation (A.109)] is not normalized and must be renormalized. The energy of a truncated perturbation expansion [equation (A.110)] is not variational, and it may be possible to calculate energies lower than experimental. ... 

It should be noted that (4.28) is only an approximation for the nuclear wave function. The perturbation terms (4.36) will mix into the nuclear wave function small contributions from harmonic-oscillator functions with quantum numbers other than v. These anharmonicity corrections to the vibrational wave function will add further to the probability of transitions with At) > 1. 

Za is the charge on nucleus A located at point ta, and plE) is the total electronic density, (r ) (r ), at each point in space E, and T (r ) is the molecular wave function. The perturbation caused by the unit test charge is not considered rather it is simply a hypothetical probe to obtain the relative energy of interaction at points surrounding a molecule. 

The Breit—Pauli spin-orbit Hamiltonian is very useful for organic molecules when the matrix elements of are computed from the nonrelativistic wave functions using perturbation theory or response theory, but it often overestimates the magnitude of spin-orbit splitting. It also suffers from... 

The first two terms in Eq. 5.15 provide a difference of the polarization energy of the QM/EFP system in the excited and ground electronic states, while the last term is the leading correction to the interaction of the ground-state-optimized induced dipoles with the excited state wave function. The perturbative treatment of the response of the polarizable environment as in Eqs. 5.14 and 5.15 corresponds to the "method 2 from Ref. [56]. This approach has been used for calculations discussed in Section 5.6. 

The peiturbational estimate mentioned relies on the calculation of the weight of the determinant based on the first-order correction to the wave function in perturbation theory (p. 245). In such an estimate, the denominato- contains the excitation eneigy evaluated as the difference in rxbital eneigies between the Hartree-Fock determinant and the one in questirm. In the numeialor, there is a respective matrix elranent of the Hamiltonian calculated with the help of the known Slater-Condon rules (see Appendix M available at bottoite.elsevier.com/978-0-444-59436-5, p. el09>. 

Oq is the simple product of the two monomers wave functions. The perturbation oper-... 

Molecular theory of the VCD by Stephens involves the determination of wave-function parameters perturbed with respect to the nuclear displacements and the magnetic perturbation. It means, it requires solution of 3V-I-3 response equations N is the number of atoms in the system). Theory of the VCD spectra is constructed within the following assumptions of (a) Born-Oppenheimer approximation (BO) (15.1), in which total wavefunction is a product of electronic and nuclear wave functions for an electronic reference state gO and vibrational state gl ... 

The time-dependent theory of spectroscopy bridges this gap. This approach has received less attention than the traditional time-independent view of spectroscopy, but since 1980, it has been very successfully applied to the field of coordination chemistry.The intrinsic time dependence of external perturbations, for example oscillating laser fields used in electronic spectroscopy, is also expKdtly treated by modern computational methods such as time-dependent density functional theory, a promising approach to the efficient calculation of electronic spectra and exdted-state structures not based on adjustable parameters, as described in Chapter 2.40. In contrast, the time-dependent theory of spectroscopy outlined in the following often relies on parameters obtained by adjusting a calculated spectrum to the experimental data. It provides a unified approach for several spectroscopic techniques and leads to intuitive physical pictures often qualitatively related to classical dynamics. The concepts at its core, time-dependent wave functions (wave packets) and autocorrelation functions, can be measured with femtosecond (fs) techniques, which often illustrate concepts very similar to those presented in the following for the analysis of steady-state spectra. The time-dependent approach therefore unifies spectroscopic... 

As a result of the heterogeneous distributions of electrical charge and chemical functionalities present on the surface of most proteins, the adsorption of proteins to solid substrates of differing surface properties may produce different molecular orientations. The optical thickness of protein layers can be evaluated by using el-lipsometry, surface plasmon resonance, or guided wave perturbation. When a chro-mophore of the protein is used as a probe, polarized spectroscopic techniques are found to be applicable to measure the tilt angle of the heme moiety in protein films. Macdonald and Smith used SERRS... 

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