Wave functions equations, perturbed

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. ... 

In this section we outline the fundamental differences in molecular properties evaluated using the EOM-CC and CCLR approaches. Although our exposition will make use of a frequency-independent (static-field) formalism, the practical extension of the final equations to frequency-dependent perturbations is trivial, requiring only the insertion of appropriate field frequencies in the perturbed wave function equations. 

The scheme for calculating polarizabilities in D dimensions follows the calculations for three dimensions presented above. The perturbation expressions are applicable independent of the dimension of the system and we can therefore use equation (13) for the polarizability. In analogy with three dimensions the unperturbed wave function (equation (14), [4]) is (for D > 2)... 

Reduced Resolvent or the Almosf Inverse of MBPT Machinery Part 1 Energy Equation MBPT Machinery Part 2 Wave Function Equation BriUouin-Wigner Perturbation Theory Rayleigh-Schrodinger Perturbation Theory... 

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 ... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

The expression for the force on the nuclei, Eq. (89), has the same form as the BO force Eq. (16), but the wave function here is the time-dependent one. As can be shown by perturbation theory, in the limit that the nuclei move very slowly compared to the electrons, and if only one electronic state is involved, the two expressions for the wave function become equivalent. This can be shown by comparing the time-independent equation for the eigenfunction of H i at time t... 

Importantly for direct dynamics calculations, analytic gradients for MCSCF methods [124-126] are available in many standard quantum chemistiy packages. This is a big advantage as numerical gradients require many evaluations of the wave function. The evaluation of the non-Hellmann-Feynman forces is the major effort, and requires the solution of what are termed the coupled-perturbed MCSCF (CP-MCSCF) equations. The large memory requirements of these equations can be bypassed if a direct method is used [233]. Modem computer architectures and codes then make the evaluation of first and second derivatives relatively straightforward in this theoretical framework. 

The solutions for the unperturbed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrodinger equation is... 

These are zero-, first-, second-, th-order perturbation equations. The zero-order equation is just the Schodinger equation for the unperturbed problem. The first-order equation contains two unknowns, the first-order correction to the energy, Wi, and the first-order correction to the wave function, 4< i. The th-order energy correction can be calculated by multiplying from the left by 4>o and Integrating, and using the turnover rule ( o Ho, ) = (, Ho o)... 

From this it would appear that the ( - l)th-order wave function is required for calculating the th-order energy. However, by using the turnover rule and the nth and lower-order perturbation equations (4.32), it can be shown that knowledge of the nth-order wave function actually allows a calculation of the (2n-i-l)th-order energy. 

Up to this point we are still dealing with undetermined quantities, energy and wave funetion corrections at each order. The first-order equation is one equation with two unknowns. Since the solutions to the unperturbed Schrddinger equation generates a complete set of functions, the unknown first-order correction to the wave function can be expanded in these functions. This is known as Rayleigh-Schrddinger perturbation theory, and the equation in (4.32) becomes... 

The use of Cl methods has been declining in recent years, to the profit of MP and especially CC methods. It is now recognized that size extensivity is important for obtaining accurate results. Excited states, however, are somewhat difficult to treat by perturbation or coupled cluster methods, and Cl or MCSCF based methods have been the prefen ed methods here. More recently propagator or equation of motion (Section 10.9) methods have been developed for coupled cluster wave functions, which allows calculation of exited state properties. 

Just as the variational condition for an HF wave function can be formulated either as a matrix equation or in terms of orbital rotations (Sections 3.5 and 3.6), the CPFIF may also be viewed as a rotation of the molecular orbitals. In the absence of a perturbation the molecular orbitals make the energy stationary, i.e. the derivatives of the energy with respect to a change in the MOs are zero. This is equivalent to the statement that the off-diagonal elements of the Fock matrix between the occupied and virtual MOs are zero. 

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,... 

Moreover, if the wave function + Xxp P is used as a trial function 0, then the quantity W from equation (9.2) is equal to the second-order energy determined by perturbation theory. Any trial function 0 with parameters which reduces to -h 20o for some set of parameter values yields an approximate energy W from equation (9.2) which is no less accurate than the second-order perturbation value. 

An expression for e(k) in the case of a Fermi gas of free electrons can be obtained by considering the effect of an introduced point charge potential, small enough so the arguments of perturbation theory are valid. In the absence of this potential, the electronic wave functions are plane waves V 1/2exp(ik r), where V is the volume of the system, and the electron density is uniform. The point charge potential is screened by the electrons, so that the potential felt by an electron, O, is due to the point charge and to the other electrons, whose wave functions are distorted from plane waves. The electron density and the potential are related by the Poisson equation,... 

ABBA molecules, 631-633 HCCS radical, 633-640 perturbative handling, 641-646 theoretical principles, 625-633 Hamiltonian equation, 626-628 vibronic problem, 628-631 Thouless determinantal wave function, electron nuclear dynamics (END) ... 

In this diabatic Schrodinger equation, the only terms that couple the nuclear wave functions Xd(R-/v) are the elements of the W RjJ and zd q%) matrices. The —(fi2/2p)W i(Rx) matrix does not have poles at conical intersection geometries [as opposed to W(2 ad(R>.) and furthermore it only appears as an additive term to the diabatic energy matrix cd(q>.) and does not increase the computational effort for the solution of Eq. (55). Since the neglected gradient term is expected to be small, it can be reintroduced as a first-order perturbation afterward, if desired. 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

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. 

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