1 Rayleigh-Schrödinger Perturbation Theory

Transcription

1 1 Rayleigh-Schrödinger Perturbation Theory All perturbative techniques depend upon a few simple assumptions. The first of these is that we have a mathematical expression for a physical quantity for which we are unable to obtain an exact solution. The next assumption is that this physical quantity may be broken down into a part which can be solved exactly and a troublesome part which has no analytic solution. This perturbation is assumed to be relatively small in comparison to the soluble portion of our problem. In our analysis, we will also assume that the eigenvalues of our exactly soluble part of the problem are non-degenerate. In RSPT the equation we wish to solve is given by ĤΨ n = E n Ψ n, (1) where Ĥ represents the Hamiltonian for our system of interest and Ψ n is an exact eigenfunction of the Hamiltonian. In order to be able to apply RSPT to this problem, we must be able to break down our Hamiltonian into two Hermitian parts, one which is soluble and the other which is not: Ĥ = Ĥo + λ ˆV. (2) Ĥ 0 is known as the unperturbed Hamiltonian or the zeroth order Hamiltonian, while ˆV is termed the perturbation. Here we have introduced the parameter λ, which is assumed only to be a real term with a value between 0 and 1. The utility of this parameter requires some motivation. If λ is taken to be zero, equation (??) reduces to the zeroth order equation, Ĥ o Ψ (0) n = E (0) n Ψ (0) n. (3) As λ is allowed to increase in value, a perturbation is introduced to both the energy and wavefunction of equation (??): E n = E n (0) + E n (4) Ψ n = Ψ (0) n + Ψ n (5) Clearly, these expressions for E and Ψ are dependent upon the parameter λ. With this in mind, we can write an expansion for each in terms of an expansion in powers of λ. E n = E n (0) + λe n (1) + λ 2 E n (2) + λ 3 E n (3) + (6) Ψ n = Ψ (0) n + λψ (1) n + λ 2 Ψ (2) n + λ 3 Ψ (3) n + (7) 2

2 These two equations are merely power series expansions which have employed the following simplifications. E n (k) = 1 d k E n k! dλ k (8) Ψ (k) n = 1 k Ψ n k! λ k (9) We are free to constrain the higher order corrections to Ψ (0) n condition that Ψ (0) n Ψ (m) n = δ m0 (10) is normalized, we have what is known as intermediate nor- As long as Ψ (0) n malization: with the Ψ (0) n Ψ n = 1 (11) If the expressions for E n and Ψ n in equations (6) and (7) are introduced to (??) and coefficients of like powers of λ on each side of the equation are set equal to each other, we get an infinite number of equations of the form Ĥ 0 Ψ (0) n = E n (0) Ψ (0) n (12) Ĥ 0 Ψ (1) n + ˆV Ψ (0) n = E n (0) Ψ (1) n + E n (1) Ψ (0) n (13) Ĥ 0 Ψ (2) n + ˆV Ψ (1) n = E n (0) Ψ (2) n + E n (1) Ψ (1) n + E n (m) Ψ (0) n (14) Ĥ 0 Ψ (3) n + ˆV Ψ (2) n = E n (0) Ψ (3) n + E n (1) Ψ (2) n + E n (2) Ψ (1) n + E n (3) Ψ (0) n (15) Taking advantage of the orthogonality relation (??) we obtain the interesting series of equations E n (0) = Ψ (0) n Ĥ0 Ψ (0) n (16) E n (1) = Ψ (0) n ˆV Ψ (0) n (17) E n (2) = Ψ (0) n ˆV Ψ (1) n (18) E n (3) = Ψ (0) n ˆV Ψ (2) n (19) E (m). n = Ψ (0) n ˆV Ψ (m 1) n (20) Clearly, if we wish to solve for the m th order perturbation to the energy, we must find a way to solve for Ψ (m 1) n. However, under certain conditions E n (2m) and E (2m+1) n can be determined from Ψ (m) n. 2 2 P. O. Lödin, J. Math Phys., 6, 1341, (1965). 3

3 If we return to our original assumptions about the form of the RSPT Hamiltonian, we see that H 0 is an hermitian operator, and has a set of nondegenerate solutions which are orthogonal and form a complete space. Since Ψ (0) n is one of these solutions, any vector orthogonal to it may be expressed as a linear combination of all the other solutions to the eigenvalue equation, { Ψ (0) n }: Ψ (m) n = C (m) n,l Ψ (0) l (21) l where C (m) n,l = Ψ (0) l Ψ (m) n (22) For m = 1 the C n,l s may be obtained with only the zeroth order solutions. Left multiplication of equation (13) by Ψ (0) l yields and so (E (0) n E (0) l ) Ψ (0) l Ψ (1) n = Ψ (0) l ˆV Ψ (0) n (23) C (1) n,l = Ψ(0) (E (0) n l ˆV Ψ (0) n E (0) l ) (24) Expansions for the higher order corrections to Ψ n may be obtained in a similar manner with increasingly complicated expressions for the expansion coefficients. These expressions for the perturbed wavefunction lead directly to the perturbed energies via equation (20). At this point it is interesting to note that we have obtained expressions for E n through infinite levels of perturbation without saying anything about the nature of Ĥ 0 or ˆV. One can envision a great number of ways in which the Hamiltonian for a system of particles could be partitioned. Obviously, for a given physical situation, certain partitionings will yield more accurate predictions than others, and certain partitionings will lend a more logical and intuitive structure to the RSPT equations. For quantum chemists, the first guess at the exact wavefunction for a molecular system is the Hartree-Fock wavefunction. From this first guess, getting the exact answer involves including all the electron correlation via a full CI. Within this logical framework, treating electron correlation as a perturbation on the HF solution has an intuitive appeal. This appealing partitioning of the Hamiltonian forms the basis for Møller-Plesset perturbation theory. 4

4 2 Møller-Plesset Perturbation Theory Møller-Plesset perturbation theory (MPPT) 3, which is a particular formulation of many body perturbation theory (MBPT), takes Ĥ0 to be the sum of the one-electron Fock operators, and treats electron correlation as the perturbation to the zeroth-order Hamiltonian. This formulation of PT is the one most commonly used by quantum chemists. One of MPPT s distinguishing features is size extensivity: the predicted energy for every order of perturbation in MPPT scales with the number of non-interacting particles in the system. This aspect of MPPT contrasts it to configuration interaction methods which are not size extensive. Size extensivity is an important issue when comparing systems with differing numbers of electrons and when treating infinite systems such as crystal lattices. Also in contrast to CI methods, however, perturbative treatment of the electron correlation energy does not give a total electronic energy which is variational. The formal expansion of the MPPT partitioned Hamiltonian 4 may be written as Ĥ = Ĥ0 + ˆV (25) where Ĥ 0 = i f(i) = i h(i) + V HF (i) (26) and ˆV = i<j(r 1 ij V HF ) (27) by Recall that a matrix element of the Hartree-Fock potential term is given V HF pq = b pb qb (28) where the sum over b includes all occupied spin orbitals, and the p and q indecies correspond to the pth and qth HF spin-orbital. Our zeroth order wavefunction, then, is simply the HF wavefunction, and the zeroth-order energy is the sum of the orbital energies of the occupied orbitals {ɛ a }. Equation 3 C. Møller and M. S. Plesset, Phys. Rev., 46, 618, (1934). 4 A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, 1st Ed., revised (McGraw- Hill, New York, 1989). 5

5 (11) tells us that the first-order energy correction is given by making the total first order energy E (1) 0 = Ψ (0) 0 ( 1 r 12 ˆV HF ) Ψ (0) 0 (29) E n = E n (0) + E n (1) = a ɛ a 1 ab ab (30) 2 ab which is just the HF energy. The first correction to the HF energy does not come until after first-order. Second-order MPPT, or MP2, is the method which is most widely used by quantum chemists. Higher order perturbation expansions become significantly more computationally intensive, but do not perform as well as other methods of similar or lesser computational expense. The only new information required to obtain the MP2 energy is the first order wave function. In our investigation of RSPT we declared the higher order contributions to our total electronic wavefunction to be orthogonal to Ψ (0) n. One convenient set of wavefunctions which fits this constraint is the set of determinants which represent excitations from the occupied χ i s in Ψ (0) n to spin orbitals which are unoccupied in the reference wavefunction. Inclusion of all the HF solutions in the first order wavefunction, however, turns out to be unnecessary. Slater s rules, when applied to the second order energy expression, dictate that only doubly excited determinants will have non-zero contributions to the MP2 energy. The first order wavefunction may be expanded as Ψ (1) n = C (1) n,abrs Ψrs ab (31) where Ψ rs ab represents a wavefunction which has electrons excited from spin orbitals a and b (occupied in Ψ (0) n ) into spin orbitals r and s (unoccupied in Ψ (0) 0 ), respectively. The coefficients C abrs are determined by the equation C (1) n,abrs = Ψrs ab Ψ (1) n = Ψ rs ab Ψ (0) (32) This wave-function may then be placed in the second order energy expression to give E (2) 0 = n Ψ (0) 0 1 r 12 Ψ rs ab 2 (33) 6

6 = = 1 4 abrs ab rs 2 (34) ab rs 2 (35) So far our treatment has been solely in terms of spin orbitals, but, if we are utilizing a restricted Hartree-Fock reference wavefunction, and we are only considering closed shell systems, then our energy expression becomes a great deal simpler. If we now consider the second order energy correction in terms of spatial orbitals for an N electron system E (2) 0 = 2 N 2 abrs ab rs rs ab N 2 abrs ab rs rs ba (36) where a, b, r and s each now signify spatial orbitals. Expressions for the higher-order energies may derived in a similar fashion. The actual derivation, however, involves copious amounts of tedious algebra. Alternate methods of deriving the expressions for MBPT energies have been suggested, including a diagrammatic technique first proposed by J. Goldstone. Such techniques often achieve simple expressions for algebraicly complicated terms, and, for those well acquainted with them, can serve as an interpretive tool which allows for extension to higher orders of approximation with greater facility than more obvious methods. The third and fourth order Møller-Plesset perturbation theory (MP3, MP4) are also commonly employed by quantum chemists. The third-order energy is given by D ˆV E n (3) 0s ( = ˆV st ˆV 00 δ st ) ˆV t0 (37) (E 0 E s )(E 0 E t ) st where the summation is held over the set of all doubly excited determinants, D, and the 0 index indicates Ψ (0) n, the zeroth-order wavefunciton. It is interesting to note that the third order energy still only involves double excitations from the reference wavefunction. The fourth order energy is given by the expression E (4) n = D ˆV 0s ˆVs0 ˆV0t ˆVt0 st (E 0 E s )(E 0 E t ) 2 7

7 + D su SDT Q t ˆV 0s ( ˆV st ˆV 00 δ st )( ˆV tu ˆV 00 δ tu ) ˆV t0 (E 0 E s )(E 0 E t )(E 0 E u ) (38) where the second sum over t is over the set of singly, doubly, triply and quadruplely excited determinants. The step from third-order to fourth order is a very expensive one, but may be made less so by omitting the triple excitations. Such an approximation does not destroy the size extensivity, but the results are no longer exact through fourth order, except for a collection of systems with two or fewer electrons. 3 References C. Møller and M. S. Plesset, Phys. Rev., 46, 618, (1934). R. Krishnan and J. A. Pople, Int. Journ. Quantum Chem., 14, 91 (1978). A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, 1st Ed., revised (McGraw-Hill, New York, 1989). Eugen Merzbacher, Quantum Mechanics, 2nd Ed., (John Wiley and Sons, New York, 1970). David Park, Introduction to the Quantum Theory, 3rd Ed., (McGraw-Hill, Inc., New York, 1992). D. A. McQuarrie, Quantum Chemistry (University Science Books, Mill Valley, CA, 1983). W. J. Hehre, L. Radom, P. v. R. Schleyer and J. A. Pople, Ab Initio Molecular Orbital Theory, (John Wiley and Sons, New York, 1986). 8