Perturbation theory first order effects

In the early history of high resolution NMR, the theory was developed by use of perturbation theory. First-order perturbation theory was able to explain certain spectra, but second-order perturbation theory was needed for other cases, including the AB system. Spectra amenable to a first-order perturbation treatment give very simple spectral patterns ( first-order spectra), as described in this section. More complex spectra are said to arise from second-order effects. ... 

Equation 2.24 can be thought of as having been derived from Equation 2.25 by adding the third term on the left hand side of Equation 2.24 as a perturbation. In first order quantum mechanical perturbation theory (see any introductory quantum text), the perturbation on the ground state of Equation 2.25 is obtained by averaging the perturbation over the ground state wave function of Equation 2.25. The effect of this... 

The plot of the direct SSC reveals that the value of the direct SSC does not depend on the molecular geometry. This fact demonstrates that this is mainly a first-order effect, thus justifying the use of first-order perturbation theory for its description. Moreover, it becomes obvious that the SSC contribution to the ZFS is significant and cannot be neglected in a quantitative description. 

In M0ller-Plesset theory, first-order perturbation theory does not improve on the HF energy because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order perturbation theory can be useful for estimating energetic effects associated with operators that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly difficult to self-consistently optimize wavefunctions for systems where these tenns are explicitly included in the Hamiltonian, but an estimate of their energetic contributions may be had from simple first-order perturbation theory, since that energy is computed simply by taking the expectation values of the operators over the much more easily obtained HF wave functions. 

We will later consider the approximation that affects the transition from Eq. (4.4) to Eq. (4.6) in detail. But this result would often be referred to as first-order perturbation theory for the effects of - see Section 5.3, p. 105 - and we will sometimes refer to this result as the van der Waals approximation. The additivity of the two contributions of Eq. (4.1) is consistent with this form, in view of the thermodynamic relation pdpi = dp (constant T). It may be worthwhile to reconsider Exercise 3.5, p. 39. The nominal temperature independence of the last term of Eq. (4.6), is also suggestive. Notice, however, that the last term of Eq. (4.6), as an approximate correction to will depend on temperature in the general case. This temperature dependence arises generally because the averaging ((... ))i. will imply some temperature dependence. Note also that the density of the solution medium is the actual physical density associated with full interactions between all particles with the exception of the sole distinguished molecule. That solution density will typically depend on temperature at fixed pressure and composition. 

While first-order effects of purely inductive substituents on excitation energies of alternant hydrocarbons vanish, higher-order perturbation theory gives nonzero contributions. Thus, Murrell (1963), using second-order perturbation theory, derived the relation... 

Next, we consider the even weaker second-order perturbation of the three sublevels of T that is due to Recall that the analogous second-order contributions from are neglected in this approach, as they are generally believed to be small, and the first-order contributions have already been included. It is seen from inspection of Equation 3.3 that the operator can mix each of these three triplet wave functions with singlet, other triplet, and quintet wave functions. This interaction has no first-order effect on the energies -D, -Dy, and -D of the three substates, T,, T, and T, respectively, but in second order. Equation 3.14 with V = IP, they will be affected somewhat and become -D , -Dy, and -Z) . If we continue to define the zero-field splitting parameters by D = 3DJ/2 and E = (DJ - Dy )/2, they can be compared with the values observed. In Section 3.3, we noted the difficulties involved in attempts to evaluate these values accurately in this fashion, due to the very large number of states over which the summation in Equation 3.14 is necessary, and we commented on alternative methods of evaluation such as response theory. 

The traditional role of perturbation theory in reactor physics has been to estimate, with a first-order accuracy, the effect of an alteration in the reactor on its reactivity. Lately, application of perturbation theory techniques has increased significantly in both scope and volume. Two general trends characterize these developments (1) improvement of the accuracy of reactivity calculation, and (2) extension of the use of second-order perturbation theory formulations for estimating the effect of a perturbation on integral parameters other than reactivity, and to nuclear systems other than reactors. These trends reflect two special features of perturbation theory. First, it provides exact expressions for the effect of an alteration in the reactor on its reactivity. For small, and especially local alterations, these perturbation expressions are easier and cheaper to apply than other approaches. Second, second-order perturbation theory formulations can be applied with distribution functions pertaining only to the unperturbed system. 

The second term Ep differs from the corresponding LR perturbation theory, because it is not limited to first order effects in the polarization. The SCF procedure in fact allows a complete mutual relaxation of the charge distributions of A and B under the electrostatic effect of the partner (at the H-F level). 

So by symmetry the response must be an even function of one looks first for leading terms in E. in the response theory perturbation treatment. These will evidently result from first order effects of sC polarizability torques involving P2(cos9) only but for polar molecules there can also be second order effects of torques on the permanent dipoles involving Pj(cosd) - cosd as well as 2. It is here that complications of non-linearity develop as the effect of the operator d/dp on the first order perturbation fj(t) and hence from equation (10) on F Ct) must be considered unless F] (t) - F] (0) which is independent of p. ... 

A simple example of the application of (11.9.5) is a derivation of the coupled Hartree-Fock (CHF) perturbation theory, first proposed by Peng (1941) and rediscovered, in various forms and with various generalizations, on many occasions. The essence of the approach is to start from a one-determinant wavefunction, optimized in the Hartree-Fock sense in the absence of the perturbation, and to seek the necessary first-order changes in the orbitals to maintain self-consistency when the perturbation is applied. The term coupled is used to indicate that, even if the perturbation contains only one-electron operators, the HF effective field must also change and will introduce a coupling , through the electron interactions, between the perturbation and the electron density. 

We now consider the Stark effect quantitatively for linear and symmetric-top molecules. The first-order effect possible for a symmetric top is given by standard first-order perturbation theory as... 

When X or M is zero, the first-order effect vanishes, and only a quadratic effect is found, as for a linear molecule. The second-order effect calculated for a linear molecule also applies to a symmetric top when X is zero. Standard second-order perturbation theory gives for the Stark correction of the 7, M level. 

If the long-range mteraction between a pair of molecules is treated by quantum mechanical perturbation theory, then the electrostatic interactions considered in section Al.5.2.3 arise in first order, whereas induction and dispersion effects appear in second order. The multipole expansion of the induction energy in its fill generality [7, 28] is quite complex. Here we consider only explicit expressions for individual temis in the... 

The contribution of the electron to the diamagnetic susceptibility of the system can be calculated by the methods of quantum-mechanical perturbation theory, a second-order perturbation treatment being needed for the term in 3C and a first-order treatment for that in 3C". In case that the potential function in 3C° is cylindrical symmetrical about the s axis, the effect of 3C vanishes, and the contribution of the electron to the susceptibility (per mole) is given... 

It should be noted that, due to the effect of spin-orbit interaction the correct initial and final states are not exactly the pure spin states. The admixture with higher electronic states j/ may be ignored only if there exists a direct coupling between the initial and final pure spin states. Otherwise, the wave function for the initial state is obtained to first order of perturbation theory as ... 

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,... 

In this section we will discuss perturbation methods suitable for high-energy electron diffraction. For simplicity, in this section we will be concerned with only periodic structures and a transmission diffraction geometry. In the context of electron diffraction theory, the perturbation method has been extensively used and developed. Applications have been made to take into account the effects of weak beams [44, 45] inelastic scattering [46] higher-order Laue zone diffraction [47] crystal structure determination [48] and crystal structure factors refinement [38, 49]. A formal mathematical expression for the first order partial derivatives of the scattering matrix has been derived by Speer et al. [50], and a formal second order perturbation theory has been developed by Peng [22,34],... 

In a perturbation theory treatment of the total (not just electrostatic) interaction between the molecule and the point charge, QV(r) is the first-order term in the expression for the total interaction energy (which would include polarization and other effects). 

Our analysis thus far has assumed that solution of the spin Hamiltonian to first order in perturbation theory will suffice. This is often adequate, especially for spectra of organic radicals, but when coupling constants are large (greater than about 20 gauss) or when line widths are small (so that line positions can be very accurately measured) second-order effects become important. As we see from... 

In Chapter 4 (Sections 4.7 and 4.8) several examples were presented to illustrate the effects of non-coincident g- and -matrices on the ESR of transition metal complexes. Analysis of such spectra requires the introduction of a set of Eulerian angles, a, jS, and y, relating the orientations of the two coordinate systems. Here is presented a detailed description of how the spin Hamiltonian is modified, to second-order in perturbation theory, to incorporate these new parameters in a systematic way. Most of the calculations in this chapter were first executed by Janice DeGray.1 Some of the details, in the notation used here, have also been published in ref. 8. 

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