Download 1 Rayleigh-Schrödinger Perturbation Theory

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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
ĤΨn = En Ψn ,
(1)
where Ĥ represents the Hamiltonian for our system of interest and Ψn 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 + λV̂ .
(2)
Ĥ0 is known as the unperturbed Hamiltonian or the zeroth order Hamiltonian, while V̂ 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,
(0) (0)
Ĥo Ψ(0)
n = En Ψn .
(3)
As λ is allowed to increase in value, a perturbation is introduced to both the
energy and wavefunction of equation (??):
En = En(0) + ∆En
Ψn = Ψ(0)
n + ∆Ψn
(4)
(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.
En = En(0) + λEn(1) + λ2 En(2) + λ3 En(3) + · · ·
(1)
2 (2)
3 (3)
Ψn = Ψ(0)
n + λΨn + λ Ψn + λ Ψn + · · ·
2
(6)
(7)
These two equations are merely power series expansions which have employed
the following simplifications.
1 dk En
k! dλk
1 ∂ k Ψn
=
k! ∂λk
En(k) =
(8)
Ψ(k)
n
(9)
We are free to constrain the higher order corrections to Ψ(0)
n with the
condition that
(m)
(10)
hΨ(0)
n |Ψn i = δm0
As long as Ψ(0)
n is normalized, we have what is known as intermediate normalization:
hΨ(0)
(11)
n |Ψn i = 1
If the expressions for En and Ψn 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
Ĥ0 Ψ(0)
n
(0)
Ĥ0 Ψ(1)
+
V̂
Ψ
n
n
(2)
Ĥ0 Ψn + V̂ Ψ(1)
n
(3)
Ĥ0 Ψn + V̂ Ψ(2)
n
=
=
=
=
En(0) Ψ(0)
n
(1) (0)
En(0) Ψ(1)
n + En Ψn
(1) (1)
(m) (0)
En(0) Ψ(2)
n + En Ψn + En Ψn
(1) (2)
(2) (1)
(3) (0)
En(0) Ψ(3)
n + En Ψn + En Ψn + En Ψn
(12)
(13)
(14)
(15)
Taking advantage of the orthogonality relation (??) we obtain the interesting
series of equations
En(0)
En(1)
En(2)
En(3)
En(m)
=
=
=
=
..
.
=
(0)
hΨ(0)
n |Ĥ0 |Ψn i
(0)
hΨ(0)
n |V̂ |Ψn i
(1)
hΨ(0)
n |V̂ |Ψn i
(2)
hΨ(0)
n |V̂ |Ψn i
(16)
(17)
(18)
(19)
(m−1)
i
hΨ(0)
n |V̂ |Ψn
(20)
Clearly, if we wish to solve for the mth order perturbation to the energy, we
. However, under certain conditions En(2m)
must find a way to solve for Ψ(m−1)
n
2
(2m+1)
can be determined from Ψ(m)
and En
n .
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P. O. Lödin, J. Math Phys., 6, 1341, (1965).
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If we return to our original assumptions about the form of the RSPT
Hamiltonian, we see that H0 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 }:
X (m) (0)
Cn,l Ψl
(21)
=
Ψ(m)
n
l
where
(m)
(0)
Cn,l = hΨl |Ψ(m)
n i
(22)
For m = 1 the Cn,l ’s may be obtained with only the zeroth order solutions.
(0)
Left multiplication of equation (13) by hΨl | yields
(0)
(0)
(0)
(0)
(En(0) − El )hΨl |Ψ(1)
n i = hΨl |V̂ |Ψn i
and so
(1)
Cn,l =
(23)
(0)
hΨl |V̂ |Ψ(0)
n i
(0)
(0)
(En − El )
(24)
Expansions for the higher order corrections to Ψn 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 En through infinite levels of perturbation without saying anything about
the nature of Ĥ0 or V̂ . 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.
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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 Hamiltonian4 may be
written as
Ĥ = Ĥ0 + V̂
(25)
where
Ĥ0 =
X
f (i) =
X
i
h(i) + V HF (i)
(26)
i
and
V̂ =
X
−1
(rij
− V HF )
(27)
i<j
Recall that a matrix element of the Hartree-Fock potential term is given
by
VpqHF =
X
hpb||qbi
(28)
b
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 {²a }. Equation
3
C. Møller and M. S. Plesset, Phys. Rev., 46, 618, (1934).
A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, 1st Ed., revised (McGrawHill, New York, 1989).
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(11) tells us that the first-order energy correction is given by
(1)
(0)
E0 = hΨ0 |(
1
(0)
− V̂ HF )|Ψ0 i
r12
(29)
making the total first order energy
En = En(0) + En(1) =
X
²a −
a
1X
hab||abi
2 ab
(30)
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 χi ’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 =
X
(1)
Cn,abrs Ψrs
ab
(31)
a>b;r>s
where Ψrs
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
(0)
Ψ0 ), respectively. The coefficients Cabrs are determined by the equation
(1)
(1)
Cn,abrs = hΨrs
ab |Ψn i =
(0)
hΨrs
ab |Ψn i
a>b;r>s ²a + ²b − ²r − ²s
X
(32)
This wave-function may then be placed in the second order energy expression
to give
(2)
E0
(0)
2
|hΨ0 | r112 |Ψrs
ab i|
=
a>b;r>s ²a + ²b − ²r − ²s
X
6
(33)
=
|hab||rsi|2
a>b;r>s ²a + ²b − ²r − ²s
(34)
=
1X
|hab||rsi|2
4 abrs ²a + ²b − ²r − ²s
(35)
X
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
N
(2)
E0
=2
2
X
hab|rsihrs|abi
abrs ²a
+ ²b − ²r − ²s
N
−
2
X
hab|rsihrs|bai
abrs ²a
+ ²b − ²r − ²s
(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
X
V̂0s (V̂st − V̂00 δst )V̂t0
(3)
En =
(37)
st (E0 − Es )(E0 − Et )
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
En(4) = −
D
X
st
V̂0s V̂s0 V̂0t V̂t0
(E0 − Es )(E0 − Et )2
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+
D SDT
XQ V̂0s (V̂st − V̂00 δst )(V̂tu − V̂00 δtu )V̂t0
X
su
t
(E0 − Es )(E0 − Et )(E0 − Eu )
(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.
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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).
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