Second order corrections

This State for optimization and/or second-order correction Total Energy, E(Cis) = -77.8969983928 "Copying the Cisingles density for this state as the 1-particle RhoCI density. 

L Solution of the third equation gives the second-order corrections, and so on. It is shown in the standard textbooks (e.g. Eyring, Walter and Kimball, 1944) that the solutions are... 

A Perturbation Theory is developed for treating a system of n electrons in which the Hartree-Fock solution appears as the zero-order approximation. It is shown by this development that the first order correction for the energy and the charge density of the system is zero. The expression for the second order correction for the energy greatly simplifies because of the special property of the zero order solution. It is pointed out that the development of the higher order approximation involves only calculations based on a definite one-body problem. 

Starting from the second-order perturbation equation (4.32), analogous formulas can be generated for the second-order corrections. Using intermediate normalization... 

P/h can be interpreted as an effective spin density of this open shell system. Similarly to the electron binding exjvession there is no first order contribution in the correlation potential, that is, = 0, so that 5 is correct through second order. However, the second order correction in the electron correction for... 

The quantities and E are the first-order corrections io and En, the quantities and E are the second-order corrections, and so forth. If the perturbation k is small, then equations (9.19) and (9.20) converge rapidly for all values of A where 0 A 1. 

The second-order correction E to the eigenvalue E is obtained by multiplying equation (9.23) by and integrating over all space... 

The second-order correction E is obtained from equations (9.34), (9.43), and (4.51) as follows... 

The evaluation of the first- and second-order corrections to the eigenfunctions is straightforward, but tedious. Consequently, we evaluate here only the first-order correction for the ground state. According to equations (9.33), (9.43), and (4.51), this correction term is given by... 

Since the perturbation corrections due to b q and b q vanish in first order, we must evaluate the second-order corrections in order to find the influence of these perturbation terms on the nuclear energy levels. According to equation (9.34), this second-order correction is... 

It was shown above that the cubic term in the potential function for the anharmonic oscillator cannot, for reasons of symmetry, contribute to a first-order perturbation. However, if the matrix elements of = ax3 are evaluated, it is found that this term results in a second-order correction to the... 

Once a hyperfine pattern has been recognized, the line position information can be summarized by the spin Hamiltonian parameters, g and at. These parameters can be extracted from spectra by a linear least-squares fit of experimental line positions to eqn (2.3). However, for high-spin nuclei and/or large couplings, one soon finds that the lines are not evenly spaced as predicted by eqn (2.3) and second-order corrections must be made. Solving the spin Hamiltonian, eqn (2.1), to second order in perturbation theory, eqn (2.3) becomes 4... 

Some transition ions have central hyperfine splittings somewhat greater than this value, for example, for copper one typically finds Az values in the range 30-200 gauss, and so in these systems the perturbation is not so small, and one has to develop so-called second-order corrections to the analytical expression in Equation 5.12 or 5.13 that is valid only for very small perturbations. The second-order perturbation result (Hagen 1982a) for central hyperfine splitting is ... 

FIGURE 5.7 Second-order hyperfine shift in the X-band EPR of the Cu(II)-Tris complex. The thin solid line is the experimental spectrum of 1.5 mM CuS04 in 200 mM Tris-HCl buffer, pH 8.0 taken at v = 9420 MHz and T = 61 K. Tris is tris-(hydroxymethyl)aminomethane or 2-amino-2-hydroxymethyl-l,3-propanediol. The broken lines are simulations using the parameters g = 2.047, gN = 2.228, Atl = 185 gauss. In the lower trace the second-order correction has been omitted. 

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 perturbing Zeeman interaction has no elements on the diagonal, so there is no first-order correction. The second-order corrections are... 

For the second-order correction, we can recognize that the only possible normalized trial function l) orthogonal to Ei]) in this 2x2 case is... 

By substituting Eqs. (1.16) and (1.17) into the general perturbation expressions (1.5), we can write the total first- and second-order corrections in the form... 

In this case, we can label the second-order correction as an /- j correction and write (1.22) as... 

In transition metal complexes, proton hfs are normally < 20 MHz so that the corresponding second order contributions, which amount to < 10 kHz, may usually be neglected. For nitrogen ligands, however, the second order corrections produce frequency shifts up to 200 kHz. Since hf interactions of central ions can amount to several hundred megacycles, the terms in AE become very important for a correct description of the ENDOR spectra. 

Terms containing the W intermediates no longer contain a factor of The energy-independent, third-order term, Epp (oo), is a Coulomb-exchange matrix element determined by second-order corrections to the density matrix, where... 

Bigeleisen J (1949) The relative velocities of isotopic molecules. J Chem Phys 17 675-678 Bigeleisen J (1955) Statistical mechanics of isotopic systems with small quantum corrections. I. General considerations and the rule of the geometric mean. J Chem Phys 23 2264-2267 Bigeleisen J (1998) Second-order correction to the Bigeleisen-Mayer equation due to the nuclear field shift. Proc National Acad Sci 95 4808-4809... 

Second-order correction can be implemented in a similar way. Let us illustrate a simple method for the exponential law. Retaining the second-order term of Equations (17) and (19), we obtain ... 

So far, second-order corrections have only found their application for radiogenic isotopes (see a more extensive treatment in Albarede et al. 2004). The linear changes in the apparent mass bias of Nd with mass observed by Vance and Thirlwall (2002) is certainly an indication that high precision may benefit from such an elaborate scheme on at least some instruments. 

A correlation between isotopic ratios corrected for mass fractionation may reveal (i) rormded or slopping peak tops (ii) second-order fractionation effects. The necessity of a second-order correction should be established by showing that the bias left after a first-order correction still depends smoothly on the mass. 

---