K • p perturbation theory

In the analyses of conventional ZB semiconductors, we frequently assume a symmetric parabolic band for the conduction band state, and the Luttinger-Kohn Hamiltonian is used to describe the valence band states. In general, the effective Hamiltonian is derived from a k.p perturbation theory or from the theory of invariants developed by Pikus and Bir. In the latter theory, the operator form of the effective Hamiltonian can easily be constructed from symmetry consideration alone. Within this framework, the lowest two conduction bands and the upper six valence bands are described to the second order of k. The invariant forms of the Hamiltonians are written as follows [26,27] ... 

The critical 3p i-2p i spacing is expected to be independent of the nature of the chemical donor, but it differs significantly between Sica and Op. This reflects the fact that the CB minimum for the donors on P site is associated with the Xi camel s back structure. Variational calculations based on k.p perturbation theory have been performed for P-site donor and compared self-consistently with spectroscopic data [40]. The calculations are performed as a function of the ratio of a non-parabolicity parameter9 Q to the separation A between CBs Xi and X3. The authors use an anisotropy parameter p equal... 

Values of the coefficient of the spin splitting proportional to of Fi-conduction band for k [110] as obtained with the LMTO method, the p 16 X 16 Hamiltonian, and k p perturbation theory (PT). Experimental data are from Refs. [34,35] Units eVA . For GaAs, GaSb, and InP experiments only give the magnitude. For InSb also the sign was determined. [35]... 

This can be done either by perturbation theory or by solving a secular determinant, arising from (30), using the k = ko solutions as basis functions for other values of k. The former procedure is called k p perturbation theory. An excellent review has been given by Kane. The other scheme is called k p interpolation theory and has received close attention in a recent article by Van Dyke. ... 

Sarka, K., Demaison, J. Perturbation theory efi ective Hamiltonians and force constants, in Jensen, P, Bunker, P.R., editors. Computational Molecular Spectroscopy. Chichester Wiley 2000, p. 255-303. 

Because of the appearance of terms q p " (k) in the above expressions, this approach is known as q p perturbation theory. The quantifies defined in Eq. (3.44) are elements of a two-index matrix (n and n y, the diagonal matrix elements are simply the expectation value of the momentum operator in state We can... 

Andersson K, Mahnqvist P, Roos BO, Sadlej AJ, Wolinski K. Second-order perturbation theory with a CASSCF reference function. J Phys Chem. 1990 94 5483. 

Bukowski R, Sadie] J, Jeziorski B, Jankowski P, Szalewicz K, Kucharski S A, Williams H L and Rice B M 1999 Intermolecular potential of carbon dioxide dimer from symmetry-adapted perturbation theory J. Chem. Phys. 110 3785... 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

Roos, B. O., Andersson, K., Fiilscher, M. R, Malmqvist, P.-A., Serrano-Andres, L., 1996, Multiconfigurational Perturbation Theory Apphcations in Electronic Spectroscopy in Advances in Chemical PhysicsXCIII, Prigogine,... 

Roos BO, Andersson K, Fulscher MP, Malmqvist PA, Serrano-Andres L, Pierloot K, Merchan M (1996) Multiconfigurational perturbation theory applications in electronic spectroscopy. In Pri-gogine I Rice SA (eds) New methods in computational quantum mechanics, Vol. 93 of Advances in Chemical Physics, Wiley, New York, p 219... 

Now, let us discuss the rate equations embodied in eq.(74). To do this, there is need of a statistical analysis. If the system is kept coupled to a thermostat at absolute temperature T, and assuming that w(i - >if) contains effects to all orders in perturbation theory, the rate of this unimolecular process per unit (state) reactant concentration k + is obtained after summation over the if-index is carried out with Boltzman weight factors p(if,T) ... 

In the standard choice BHF the self-consistency requirement (5) is restricted to hole states (k < kF, the Fermi momentum) only, while the free spectrum is kept for particle states k > kF- The resulting gap in the s.p. spectrum at k = kF is avoided in the continuous-choice BHF (ccBHF), where Eq. (5) is used for both hole and particle states. The continuous choice for the s.p. spectrum is closer in spirit to many-body Green s function perturbation theory (see below). Moreover, recent results indicate [6, 7] that the contribution of higher-order terms in the hole-line expansion is considerably smaller if the continuous choice is used. 

R. K. Chaudhuii, K. F Freed, G. Hose, P. Piecuch, K. Kowalski, M. Woch, S. Chattopadhyay, D. Mukherjee, Z. Rolik, A. Szabados, G. Toth, and P. R. Surjan, Comparison of low-order multireference many-body perturbation theories. J. Chem. Phys. 122, 134105 (2005). 

K. F. Freed, Tests and applications of complete model space quasidegenerate many-body perturbation theory for molecules, in Many-Body Methods in Quantum Chemistry (U. Kaldor, ed.), Springer, Berlin, 1989, p. 1. 

K. Andersson, P. A. Malmqvist, and B. O. Roos,/. Chem. Phys., 96,1218 (1992). 2nd-Order Perturbation Theory with a Complete Active Space Self-Consistent Field Reference Function. 

K.E. Riley, P. Hobza, Investigations into the nature of halogen bonding including symmetry adapted perturbation theory analyses. J. Chem. Theor. Comput. 4, 232-242 (2008)... 

B. Jeziorski, K. Szalewicz, Intermolecular Interactions by Perturbation Theory, in Encyclopedia of Computational Chemistry, ed. by P. von Rague Schleyer, N.L. Allinger (Wiley, Chichester, 1998)... 

L. A. Curtiss, P. C. Redfern and K. Raghavachari, Gaussian-4 theory using reduced order perturbation theory, J. Chem. Phys. 127, 124105 (2007). 

Next, we consider the spin-orbit coupling, hi WZ structure, one may apply a unitary transformation, which diagonalises at the T point, to the k.p Hamiltonian and then use a perturbation theory for the states close enough to the T point, as described by Bir and Pikus [3]. This leads to the following closed expressions for the hole masses ... 

Jeziorski B, Szalewicz K (1998) Intermolecular interactions by perturbation theory. In von Rague Schleyer P, Allinger NL, Clark T, Gasteiger J, Kollman PA, Schaefer III HF, Schreiner PR (eds) Encyclopedia of computational chemistry, vol 2. Wiley, New York, ppl376-1398... 

Bukowski R, Jankowski P, Jeziorski B, Jeziorska M, Kucharski SA, Moszynski R, Rybak S, Szalewicz K, Williams HL, Wormer PES (1996) SAPT96 An ab initio program for manybody symmetry-adapted perturbation theory calculations of intermolecular interaction energies. University of Delaware and University of Warsaw... 

Merchan M, Serrano-Andres L, Fiilscher MP, Roos BO (1999) Multiconfigurational Perturbation Theory Applied to Excited States of Organic Compounds. In Hkao K (ed) Recent Advances in Multireference Methods, World Scientific, Singapore, p 161. 

Andersson, K. Mahnqvist, P.-A. Roos, B. O. Second-order perturbation theory with a complete active space self-consistent field reference function, J. Chem. Phys. 1992, 96, 1218-1226. 

We do this by using the k p method, (called k-dol-p), which is based upon the perturbation theory of Eq. (1-14). In this method, energy is calculated near a band maximum or minimum by considering the wave number (measured from the extremum) as a perturbation. (The method is described in many solid state texts, such as Kittel, 1963, p. 186, or Harrison, 1970, p. 140.) The method was used for a study of effective masses by Cardona (1963, 1965). It was also usetl in the more extensive study by Lawaetz (1971) referred to in the discussion of heavy-hole bands. We shall discuss here only the conduction band and the light-hole band where the effects of interaction are great. 

The terms omitted are of order (77,y /7,-j - Hjj ) -, they are neglected in lowest-order perturbation theory. Thus, if a p state on a chlorine ion in KCl is coupled to an s state on its neighboring potassium ion by a matrix element /7,y (which we will relate to K,p ) and the difference in energy between the two states Hjj - H,-,- = is large compared to the coupling, the probability that an electron in the perturbed state lies on that potassium ion is... 

In examining how changes in the electron states caused by the pseiidopotential change the total energy of the electron gas, it is best not to use the Fermi-Thomas approximation, used in Section 16-P but to compute the energy of the electrons directly by applying perturbation theory. For a particular electron of wave number k, Eq, (1-14) directly gives... 

T. J. Lee, A. P. Rendell, K. G. Dyall, and D. Jayatilaka,/. Chem. Phys., 100, 7400 (1994). Open-Shell Restricted Hartree-Fock Perturbation Theory Some Considerations and Comparisons. 

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