Perturbation coefficient , first-order

If the calculation of interest involves a functional that does not include exact exchange, the KS potential will be independent of the perturbation to first order except through the term if1 and. as noted earlier, (A(0-B(0)) will have a very simple form. Any if1 coefficient will then be trivial to calculate through... 

The result is very similar to that of the well-known perturbation theory. The appearance of the quantities lEk rests on the non-orthogonality of the function On the other hand there is a certain similarity in form to the energy perturbation in first order 2Ek is a linear combination with the group characters as coefficients and with the normalisation factor... 

The expansion coefficients determine the first-order correction to the perturbed wave function (eq. (4.35)), and they can be calculated for the known unperturbed wave functions and energies. The coefficient in front of 4>o for 4 i cannot be determined from the above formula, but the assumption of intermediate normalization (eq. (4.30)) makes Co = 0. 

Equation (10.42) are the first-order Coupled Perturbed Hartree-Fock (CPHF) equations." The perturbed MO coefficients are given in terms of unperturbed quantities and the first-order Fock, Lagrange (a) and overlap matrices. The F term is given as (eq. (3.52)). 

It is worth mentioning here that the sign of r x) has had a significant impact on construction of monotone schemes. One way of providing a second-order approximation and taking care of this sign is connected with a monotone scheme with one-sided first difference derivatives for the equation with perturbed coefficients... 

An instructive description of the first-order perturbation treatment of the quadrupole interaction in Ni has been given by Travis and Spijkerman [3]. These authors also show in graphical form the quadrupole-spectrum line positions and the quadrupole-spectrum as a function of the asymmetry parameter r/ they give eigenvector coefficients and show the orientation dependence of the quadrupole-spectrum line intensities for a single crystal of a Ni compound. The reader is also referred to the article by Dunlap [15] about electric quadrupole interaction, in general. 

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,... 

The determination of the coefficients Cay is not necessary for finding the first-order perturbation corrections to the eigenvalues, but is required to obtain the correct zero-order eigenfunctions and their first-order corrections. The coefficients Cay for each value of a (a = 1,2,. .., g ) are obtained by substituting the value found for from the secular equation (9.65) into the set of simultaneous equations (9.64) and solving for the coefficients c 2, , in terms of c i. The normalization condition (9.57) is then used to determine Ca -This procedure uniquely determines the complete set of coefficients Cay (a, y = 1,2, gn) because we have assumed that all the roots are different. 

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)... 

The coefficient Cb ,(t) is then obtained by a standard first-order perturbation calculation which takes into account the initial conditions defined by Eq. (7). This gives the transition probability per unit time from the initial state Xav l a to the isoenergetic continuum of states Xbw l b in the form ... 

In the present variation-perturbation calculations the first order corrections were expanded in 600-term ECG basis defined in equations (15) and (16). The components of the polarizability were computed from equation (11) using the optimized The optimization was performed separately for each component and intemuclear distance. The values of aj, (co) are arithmetic sums of the plus and minus components (equation (12)) computed from two separate first-order corrections. For a given component v (either or ), and are expanded in the same basis but, because they are solutions to two different equations (equation (9)) they differ in the linear expansion coefficients. The computed components of the static polarizability an(/ ) and a R) are drawn in Fig. 2 and their numerical values at selected intemuclear distances are listed in Table 1. 

Before closing this chapter, it is important to emphasize the context in which the transition rate expressions obtained here are most commonly used. The perturbative approach used in the above development gives rise to various contributions to the overall rate coefficient for transitions from an initial state i to a final state i these contributions include the electric dipole, magnetic dipole, and electric quadrupole first order tenns as well contributions arising from second (and higher) order terms in the perturbation solution. 

By straightforward extension of the calculation of the first order perturbation, we obtain the following difference equation for the effective diffusion coefficient Dm ... 

Because the pulse is over at t = 0, a , the preparation coefficients , are given, in first order perturbation theory, as [9, 10],... 

Coupling also influences the relative intensity of the absorptions by mixing vibrational excitations of the two molecules (first-order perturbation theory gives a mixing coefficient of C/A). If M denotes the hypothetical intrinsic intensity ratio of the individual molecules (a function of IR polarization), and r denotes the observed intensity ratio, the following relationship allows more sensitive determination of small coupling constants. 

The basis set representing the first order perturbed orbitals should also be chosen such that it satisfies the imposed finite boundary conditions and can be represented by a form like Equation (36) with the STOs having different sets of linear variation parameters and preassigned exponents. The coefficients of the perturbed functions are determined through the optimization of a standard variational functional with respect to, the total wavefunction . The frequency dependent response properties of the systems are analyzed by considering a time-averaged functional [155]... 

Equation (2.10) is a set of first-order differential equations which describe the evolution of the molecular system under the external perturbation h(t). It is valid for time-independent as well as time-dependent perturbations and must be solved subject to the initial conditions a (0) = 1 and oa j(0) = 0 if the molecule is initially in state F ). The time dependence of the coefficients aa(t) together with the stationary basis functions Fa(Q,q) describe completely the state of the molecule at each instant t. When the perturbation is switched off, the coefficients aa(t) become constant again. 

Under the assumption that the coupling elements daa> are very small, Equation (2.16) may be solved by first-order perturbation theory the coefficients aa/ (t) on the right-hand side are replaced by their initial values at t = 0. The evolution of each final state (/ i) is then governed by the (uncoupled) equation... 

Using expansion (16.2) for the wavepacket in terms of the stationary wavefunctions we can derive a set of coupled equations for the expansion coefficients au(t) similar to (2.16). In the limit of first-order perturbation theory [see Equation (2.17)] the time dependence of each coefficient is then given by... 

By this, the expansion coefficients uffl are themselves of the 0-eth order in A. The restriction l / k indicates that the correction is orthogonal to the unperturbed vector. In order to get the corrections to the /c-th vector, we find the scalar product of the perturbed Schrodinger equation for it written with explicit powers of A with one of the eigenvectors of the unperturbed problem p (j k). For the first order in A we get ... 

Because of the spin-orbit selection rules, only triplet zeroth-order states contribute to the first-order perturbation correction of a singlet wave function. In Rayleigh-Schro dinger perturbation theory, the expansion coefficient a of a triplet zeroth-order state (3spin-orbit matrix element with the electronic ground state (in the numerator) and its energy difference with respect to the latter (in the denominator). 

Accordingly, the first-order spin-orbit perturbation of a triplet wave function may be written as a linear combination of unperturbed singlet, triplet, and quintet states with expansion coefficients defined in a similar way as those in Eq. [218]. 

Under these circumstances we obtain, in first-order perturbation theory, that the expansion coefficients in Eq. (2.2) are given by... 

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