First-order perturbation equations

The calculation is classical, provided that we remember to treat the quasi-degenerate combinations of ncc with 7tCN and n cc with it CN using first-order perturbation equations (p. 30). For example, the 1-azabutadiene HOMO is ncc perturbed by nCN and 7i CN. Therefore, it lies at... 

For t > 0, a weak perturbing potential Av(rt) induces a screening potential such that AQ = Av + AQS. The first-order perturbation equations are... 

Q = L X ). The effect of anharmonicity can be accounted for by perturbation theory. For instance, the first-order perturbation equation is given by ... 

In zeroth-order e is chosen to be diagonal and the result is the ordinary field-free Hartree-Fock crystal orbital equation. As usual C(k) is complex for arbitrary k. Apart from this aspect the most important difference between tlie crystal orbital and molecular TDHF perturbation equations is the presence of tlie dC/dk term in the former. Since dC/dk is multiplied by E, field-free derivatives of C(k) with respect to k appear for the first time in the first-order perturbation equations. These field-free... 

This proves the Hylleraas variational prineiple. The last equality follows from the first-order perturbational equation, and the last inequality from the faet that is assumed to be the lowest eigenvalue of (see the variational prineiple). 

The first-order perturbation equation (p. 243) after inserting... 

The same solution is obtained when inserting Eq. (5.32) into the first-order perturbational equation, carrying out the scalar products of this equation consecutively with, m = 1, 2,..., g and solving the resulting system of equations. 

To find the conditions that (pf and must satisfy, we return to the first-order perturbation equation (12-11) but with (pf in place of rpi. ... 

This means that the first-order perturbation equation is satisfied for degenerate states only when the wavefunctions diagonalize the perturbation operator H. Therefore, our problem reduces to finding those linear combinations of and ip-j that will diagonalize H. We have already seen two ways to accomplish this in earlier chapters. One way is to construct the matrix /-/, where = and then find the unitary... 

An expression for the first-order wave function may be obtained by rearranging the first-order perturbation equation (1.28) as... 

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)... 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

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,... 

Thus, the first-order perturbation to the eigenvalue is zero. The second-order term E is evaluated using equations (9.34), (9.50), and (4.50), giving the result... 

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. 

If spin effects are neglected, the ground-state unperturbed energy level is non-degenerate and its first-order perturbation correction is given by equation (9.24) as... 

The first-order perturbation correction to the ground-state energy is obtained by evaluating equation (9.24) with (9.80) as the perturbation and (9.82) as the unperturbed eigenfunction... 

Numerical values of E > and E + for the helium atom (Z = 2) are given in Table 9.1 along with the exact value. The unperturbed energy value E l has a 37.7% error when compared with the exact value. This large inaccuracy is expected because the perturbation H in equation (9.80) is not small. When the first-order perturbation correction is included, the calculated energy has a 5.3% error, which is still large. 

Since equation (10.43) with F = 0 is already solved, we may treat V as a perturbation and solve equation (10.43) using perturbation theory. The unperturbed eigenfunctions S H q) are the eigenkets n) for the harmonic oscillator. The first-order perturbation correction to the energy as given by equation (9.24) is... 

Ed being the energy of the fast electron. To a good approximation, the effect of inelastically scattered electrons on the elastic electron wave field may be treated via a first order perturbation method. From Equation (4) we have... 

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. 

We call the Fukui function / (r) the HOMO response. Equation 24.39 is demonstrated as follows. The PhomoW is the so-called Kohn-Sham Fukui function denoted as f (r) [32]. According to the first-order perturbation theory, one has... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

There are two main ingredients that go into the semiclassical tiunover theory, which differ from the classical limit. In the latter case, a particle which has energy E > 0 crosses the barrier while if the energy is lower it is reflected. In a semiclassical theory, at any energy E there is a trarrsmission probabihty T(E) for the particle to be transmitted through the barrier. The second difference is that the bath, which is harmonic, may be treated as a qrrantum mechanical bath. Within first order perturbation theory, the equations of motion for the bath are those of a forced oscillator, and so their formally exact quantum solution is known. 

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

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