Wave function derivation

The relationship between the GSJ and is given in Table II and the values of G(J for the three cases under discussion are given in Table III. The group overlap integrals calculated using wave functions derived by... 

Thus, with the mlNln2lNl configuration we can do without the wave functions derived using the vectorial coupling of momenta of individual shells and use the functions characterized by eigenvalues of the commuting operators N, T2, Tz, L2, Lz, S2, Sz... 

The m — 6 system will again be used as an example. The guest molecules cause the mixing of the lowest (r = 0) wave function with three other wave functions derived from p = 1, p = 2, and p — 3, as described in the secular equation (14). If cx,. . ., c5 are the coefficients of the basis functions in order of increasing energy in the perturbed lowest state, we have, by perturbation theory for small a,... 

Semi-classical methods of computing otu generally involve the use of equation (IS) with wave functions derived firom model considerations. The accuracy and complexity of the calculation depend greatly on the... 

Schaefer and co-workers have presented several reports [65] about derivatives of SCF wavefunctions, including harmonic transition moments that are electrical property derivatives. They elect to give expressions directly in terms of one- and two-electron integrals. This alternative formulation of the general problem is the most immediate means of solution, but it must be done tediously, order by order. However, it has been successfully worked out to low order entirely for closed-shell, open-shell, and certain MC-SCF wave-functions. Derivatives of correlated wavefunctions may be found by the general schemes discussed at the outset of this section. 

Massa, L., Goldberg, M., Frishberg, C., Boehme, R. F. and LaPlaca, S. J. Wave functions derived by quantum modeling of the electron density from coherent X-ray diffraction beryllium metal. Phys. Rev. Lett. 55, 622-625 (1985). 

The first contribution depends upon the second-order behavior of the Hamiltonian operator and the unperturbed reference state wave function, while the second term (which will be subsequently be referred to as the relaxation contribution or relaxation term ) depends on the derivative of the wave function. This is perhaps most easily appreciated by inserting the equation for the wave-function derivative into that for the second derivative of the energy, giving... 

D. J. Grimwood, I. Bytheway and D. Jayatilaka, Wave functions derived from experiment. V. Investigation of electron densities, electrostatic potentials and electron localization functions for noncentrosymmetric crystals, J. Comp. Chem. 24, 470 83 (2003). 

According to Ref. [65], an appropriate guiding criterion to define a physically plausible exterior orbital density consists in using the functional form for the wave function derived from exact solutions to the Schrodinger equation for the hydrogen atom confined by a soft spherical box [34], i.e. ... 

The electronic Hamiltonian is diagonal in the rigged-BO basis and it remains now to study eq.(4). The equivalence E,(a aoi)= Ej(R aoi)= Ej(R) is fulfilled by the wave functions derived from the auxiliary model. By multiplying eq.(4) from the left by Y (p aoi) and integrate over the electronic coordinates we obtain... 

In contrast to variational wave functions where the first order response El — dE Q)/dQ)Q equals the expectation value of the perturbation operator with the unperturbed wave function (Hellmann-Feynman theorem) a general expression has to be used in combination with non-variational wave functions derived by differentiating all Q-dependent terms in eq. (55) ... 

Both APTs and AATs depend on the wave function derivatives with respect to nuclear displacements the computation of these terms for a molecule in solution has been already treated above. AATs depend also on the derivative of the wave function with respect to an external magnetic field. This derivatives is already known as it enters in the general definition of the nuclear shielding tensors discussed in the previous section. Once again, for its evaluation its possible to exploit the GIAO method. 

The wave function of fragment A, ll a, can either be a single determinant from HF theory or Kohn-Sham DFT, or a multiconfiguration wave function derived from complete active space self-consistent field (CASSCF) or valence bond (VB) calculations. 

The next step was sustained by the assumption that the correlation energy can be seen as the perturbation of the self-consistent-field energy, which is associated with a wave function derived for a single electronic configuration. At this point the basic methods of approximation used in quantum chemistry, namely the perturbation and variational, can be considered. 

Unfortunately, there are several misconceptions about HKS theories. We have to remember that all wave functions derived from density functional theory (DFT) are single Slater determinants and therefore represent a function that does not possess enough flexibility to mathanatically represent the true wave function. 

The resultant wave functions derived from first-order perturbation theory with 8a = - (pj + = -15.00 ev (a = -14.00 ev) are given in Table 5 and contrasted with... 

The transformed Hamiltonians that we have derived allow us to calculate intrinsic molecular properties, such as geometries and harmonic frequencies. We would like to be able to calculate response properties as well, with wave functions derived from the transformed Hamiltonian. If we used a method such as the Douglas-Kroll-Hess method, it would be tempting to simply evaluate the property using the nonrelativistic property operators and the transformed wave function. As we saw in section 15.3, the property operators can have a relativistic correction, and for properties sensitive to the environment close to the nuclei where the relativistic effects are strong, these corrections are likely to be significant. To ensure that we do not omit important effects, we must derive a transformed property operator, starting from the Dirac form of the property operator. 

Having discussed the M0ller-Plesset corrections to the wave function, we now turn our attention to the energies. From the MP2 wave function derived in the previous subsection, we may in principle calculate energies to fifth order in the fluctuation potential. We shall here be somewhat less ambitious and restrict ourselves to energies that are correct to third order in the perturbation. According to the In + 1 rule, we may calculate these corrections from the first-order wave function, derived in Section 14.2.2. 

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