First-order correction

For the partition fiinction, the contribution from Xj, which is the first-order correction in h, vanishes identically. One obtains... 

The small additions to all themiodynamic potentials are the same when expressed in temis of appropriate variables. Thus the first-order correction temi when expressed in temis of Vand P is the correction temi for the Helmlioltz free energy A ... 

L is Avagadro s constant and k is defined above. It can be seen that there are indeed two corrections to the conductivity at infinite dilution tire first corresponds to the relaxation effect, and is correct in (A2.4.72) only under the assumption of a zero ionic radius. For a finite ionic radius, a, the first tenn needs to be modified Falkenliagen [8] originally showed that simply dividing by a temr (1 -t kiTq) gives a first-order correction, and more complex corrections have been reviewed by Pitts etal [14], who show that, to a second order, the relaxation temr in (A2.4.72) should be divided by (1 + KOfiH I + KUn, . The electrophoretic effect should also... 

At finite concentration, tire settling rate is influenced by hydrodynamic interactions between tire particles. For purely repulsive particle interactions, settling is hindered. Attractive interactions encourage particles to settle as a group, which increases tire settling rate. For hard spheres, tire first-order correction to tire Stokes settling rate is given by [33]... 

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. 

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)... 

Up to this point we are still dealing with undetermined quantities, energy and wave funetion corrections at each order. The first-order equation is one equation with two unknowns. Since the solutions to the unperturbed Schrddinger equation generates a complete set of functions, the unknown first-order correction to the wave function can be expanded in these functions. This is known as Rayleigh-Schrddinger perturbation theory, and the equation in (4.32) becomes... 

The first-order correction to the wave function can be obtained by multiplying (4.32) from the left by a function other than 4>o(4 ) and integrating to give... 

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. 

The formula for the first-order correction to the wave function (eq. (4.37)) similarly only contains contributions from doubly excited determinants. Since knowledge of the first-order wave function allows calculation of the energy up to third order (In - - 1 = 3, eq. (4.34)), it is immediately clear that the third-order energy also only contains contributions from doubly excited determinants. Qualitatively speaking, the MP2 contribution describes the correlation between pairs of electrons while MP3 describes the interaction between pairs. The formula for calculating this contribution is somewhat... 

In order to achieve a high aceuraey, it would seem desirable to explicitly include terms in the wave functions which are linear in the intereleetronie distanee. This is the idea in the R12 methods developed by Kutzelnigg and co-workers. The first order correction to the HF wave funetion only involves doubly exeited determinants (eqs. (4.35) and (4.37)). In R12 methods additional terms are included which essentially are the HF determinant multiplied with faetors. 

Owing to the divergence of the K expansion near the nuclei, the mass-velocity and Darwin corrections can only be used as first-order corrections. An alternative method is to partition eq. (8.13) as in eq (8.24), which avoids the divergence near the nucleus. 

Since the first order corrections are represented by the operator W = u (V — Ea)D,liit first term of H4 is the second-order perturbation for W and the additional terms that must be taken into account to carry out a consistent calculation up to 1/c are... 

Although this lowest order approximation is used in determining the first order corrections to the distribution function, it is necessary to go to a higher order of approximation in determining the collision integral of Eq. (1-140). If we keep terms to first order in the small quantity m/M, the collision integral may be evaluated to give 28... 

We substitute (6-131) into (6-130) to obtain equations for the first order corrective terms (i.e., the differential equations for dp1jdt and chjjjdt) and obtain... 

The correction to the relaxing density matrix can be obtained without coupling it to the differential equations for the Hamiltonian equations, and therefore does not require solving coupled equations for slow and fast functions. This procedure has been successfully applied to several collisional phenomena involving both one and several active electrons, where a single TDHF state was suitable, and was observed to show excellent numerical behavior. A simple and yet useful procedure employs the first order correction F (f) = A (f) and an adaptive step size for the quadrature and propagation. The density matrix is then approximated in each interval by... 

In a recent work [36], Bak et al presented a new method to obtain first-order corrected radial couplings from MCSCF wavefnnctions and applied it to three E+ states of... 

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. 

To find the first-order correction to the eigenvalue En, we multiply equation (9.22) by the complex conjugate of and integrate over all space to obtain... 

The situation where E is degenerate requires a more complex treatment, which is presented in Section 9.5. The first-order correction is obtained by combining equations (9.30) and (9.32)... 

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... 

The matrix elements n x n) for the unperturbed harmonic oscillator are given by equations (4.50). The first-order correction term is obtained by substituting equations (9.50) and (4.50e) into (9.24), giving the result... 

However, equation (9.63) for the first-order corrections to the eigenvalues cannot be used directly at this point because the functions 0 are not known. 

Only for some values of the first-order correction term E is the secular equation (9.65) satisfied. This secular equation is of degree g in E l, giving gn roots... 

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. 

The first-order corrections to the eigenvalues are then given by... 

To obtain the first-order corrections to the eigenfunctions tpna, w multiply equation (9.62) by for k f n and integrate over all space... 

We next expand the first-order correction in terms of the complete set of unperturbed eigenfunctions... 

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