Perturbation theory, first order

The perturbation-theory first-order energy correction is (see also Problem 8.35)... 

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. ... 

The basic theoretical models to describe the interaction of ionized particles with matter were developed early in the 20th century by Bohr [1,2], Bethe [3] and Bloch [4] (BBB). These models provide the general framework to almost any consideration on the energy loss of swift particles in matter. The first two of these models are based on widely different assumptions, the Bohr description is fully classical, representing the atomic electrons by classical oscillators, while the Bethe model is based on quantum perturbation theory (first-order Born approximation). 

Long-range perturbation theory—first order... 

In the second, response theory, approach, the response of the Hartree-Fock ground state is calculated by perturbation theory. First-order perturbation theory in the fluctuation potential gives a method known as the random phase approximation (REA). The RPA linear response gives the dynamic polarizability, the quadratic response gives the first hyperpolarizability etc. One can obtain expressions for the response functions as sums over states formulae, but they are not calculated as such, rather they are calculated from coupled linear equations. RPA is equivalent to TDCPHF. 

Now consider the application of second-order perturbation theory. First, we need to represent the x—y components of the spins, Sx, Sy, Ix and Iy, in terms of their raising and lowering combinations ... 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

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... 

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. 

All other eigenvectors A4> will have eigenvalues 0(yb.b). When yb-b y we can treat yr perturbatively. Using first-order perturbation theory, we have... 

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 MCSCF provides a good first-order description covering the static electron correlation due to degeneracy problems. Dynamic electron correlation should be addressed with the MCSCF wave function as a reference. The multireference configuration interaction, or MRCI, generates excited determinants from all (or selected) determinants included in the MCSCF. The complete active space perturbation theory, second order (CASPT2) is a more economical approach. Both methods can be applied to compute excited states. 

Relativistic corrections have also been included in DFT calculations using perturbation theory, first by Herman and Skillman (1963) and later by Snijders and Baerends (Snijders 1979, Snijders and Baerends 1977, 1978). Following a non-relativistic Hartree-Fock-Slater calculation, first-order perturbation theory was used to calculate the relativistic corrections from the Breit-Pauli terms of 0(c ). Herman and Skillman applied this approach to first order only for the energies. 

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... 

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. 

One method for solving Equation 646 is to use perturbation theory. The term in the brackets in Equation 646 can be recognized as the Schroedinger equation for the RRHO approximation (Equation 6-19), and the term bs can be taken as a first-order perturbation. The first-order and higher order corrections to the energy eigenvalues to the RRHO approximation can then be computed using Perturbation Theory. 

For qualitative insight based on perturbation theory, the two lowest order energy eorreetions and the first-order wavefunetion eorreetions are undoubtedly the most usetlil. The first-order energy eorresponds to averaging the eflfeets of the perturbation over the approximate wavefunetion Xq, and ean usually be evaluated without diflfieulty. The sum of aJ, Wd ds preeisely equal to tlie expeetation value of the Hamiltonian over... 

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... 

If //j is small compared with EI we may treat EI by perturbation theory. The first-order perturbation theory fomuila takes the fonn [18, 19, 20 and 21] ... 

A very successfiil first-order perturbation theory is due to Weeks, Chandler and Andersen pair potential u r) is divided into a reference part u r) and a perturbation w r)... 

If B Bq first-order perturbation theory ean be employed to ealeulate the transition rate for EPR (at resonanee)... 

This is the final result of the first-order time-dependent perturbation theory treatment of light-indueed transitions between states i and f. 

The total rate of transitions from i to f is given, through first-order in perturbation theory, by... 

---