Chemistry 460
Spring 2015
Dr. Jean M. Standard
March 23, 2015
More on the Linear Variation Method
Accuracy of the Variation Method
Two of the reasons that the Linear Variation Method is so widely used are that it can give high accuracy results for
the approximate eigenvalues and it can be systematically improved by increasing the size of the basis set. To
understand why the Variation Method (both linear and nonlinear) gives energies that are so accurate, suppose the
approximate wavefunction φ is written
φ = cψ + δ χ ,
(1)
€
where ψ is the exact solution, χ is the error in the wavefunction, and c and δ are constants. The constants c and δ
are not arbitrary but rather an expression for€c in terms of δ can be found by requiring φ to be normalized,
€
€
φ φ
= 1
cψ + δ χ cψ + δ χ
c2 ψ ψ
+ cδ χ ψ
= 1
€
+ δc ψ χ
+ δ2 χ χ
= 1.
(2)
The assumption is that the exact wavefunction ψ and the error part of the wavefunction χ are both normalized.
Furthermore, the overlap
€ between the exact wavefunction and the error part of the wavefunction must be zero, or
else the error part of the wavefunction would contain some component of the exact wavefunction, so
χ ψ = ψ χ = 0 . Therefore,
€ Eq. (2) simplifies to
€
c 2 + δ 2 = 1.
(3)
1 − δ2 .
(4)
€
Solving for c yields,
€
c =
Substituting, the approximate wavefunction can be written as
€
φ =
1 − δ2 ψ + δ χ.
(5)
This equation shows that the parameter δ is a measure of how far the approximate wavefunction is from the exact
solution. For δ = 0 , the approximate solution
€ is exact and the error is zero. For δ = 1, the approximate
wavefunction contains no part of the exact solution (the error is 100 percent).
Now,
€ consider the expectation value of the energy for this approximate€wavefunction,
E
=
φ Hˆ φ
φ φ
=
φ Hˆ φ .
The denominator equals one since the approximate wavefunction is normalized.
€
(6)
2
Substituting Eq. (5),
E
φ Hˆ φ
=
1 − δ 2 ψ + δ χ Hˆ
=
(
= 1 − δ2
)
1 − δ2 ψ + δ χ
ψ Hˆ ψ + δ
1 − δ 2 ψ Hˆ χ
+ δ
1 − δ 2 χ Hˆ ψ
+ δ 2 χ Hˆ χ .
(7)
+ δ 2 χ Hˆ χ .
(8)
Using Hˆ ψ = E ψ , the first and third terms can be simplified,
€
E
€
(
)
1 − δ 2 ψ Hˆ χ
= 1 − δ2 E ψ ψ + δ
+ Eδ
1 − δ2
χ ψ
Normalization and orthogonality further simplify the first and third terms,
€
E
(
)
1 − δ 2 ψ Hˆ χ
= 1 − δ2 E + δ
+ δ 2 χ Hˆ χ .
(9)
The second term is simplified using the Hermitian property,
€
ψ Hˆ χ
=
Hˆ ψ χ
= E ψ χ
= 0.
(10)
Substituting,
€
E
(
)
= 1 − δ 2 E + δ 2 χ Hˆ χ .
(11)
χ Hˆ χ
(12)
Rearranging,
€
E
= E + δ2
{
}
− E .
2
This equation shows that the approximate energy equals the exact energy E plus a term that is proportional to δ .
Since δ is a measure of the €
error in the wavefunction, we see that if the wavefunction is accurate to order δ, then the
2
energy is known to order δ . For example, if δ = 0.1 , then the wavefunction is accurate to about 10%. On the
2
€
other hand, the energy is accurate to approximately 1% since its error scales like δ .
€
€
Matrix Representation of the Linear Variation Method
The secular equations of the linear variation method,
€
n
∑ ( H ki
− E Ski ) c i = 0 ,
k = 1, 2, … n ,
i=1
(13)
may be formulated in terms of a matrix equation. The n×n Hamiltonian matrix H is defined
€
"H11 H12 ! H1n %
$
'
H
H22 ! H2n '
H = $ 21
,
$ "
"
" '
$
'
#Hn1 Hn2 ! Hnn &
€
(14)
3
where the Hamiltonian matrix elements are
φ i Hˆ
φj
(15)
"S11 S12 ! S1n %
$
'
S21 S22 ! S2n '
$
S =
,
$ "
"
" '
$
'
#Sn1 Sn2 ! Snn &
(16)
Sij =
(17)
H ij =
The n×n overlap matrix S is defined
€
where the overlap matrix elements are
€
φi φ j .
The linear variation coefficients are written in the form of an n×1 vector C,
€
" c1 %
$ '
c
C = $ 2' .
$!'
$ '
#c n &
(18)
The secular equations may then be written in matrix form,
€
HC = E SC, or
{H
- E S } C = 0.
(19)
Writing out the matrix form of the secular equations yields,
€
"H11 H12 ! H1n % " c1 %
$
' $ '
$H21 H22 ! H2n ' $c 2 ' −
$ "
"
" ' $" '
$
' $ '
#Hn1 Hn2 ! Hnn & #c n &
"S11 S12
$
S
S22
E $ 21
$ "
"
$
#Sn1 Sn2
! S1n %
'
! S2n '
" '
'
! Snn &
" c1 %
"0 %
$ '
$ '
$c 2 ' = $0' .
$"'
$"'
$ '
$ '
#c n &
#0 &
(20)
When the matrices are multiplied out, the n×1 vector that is produced has as its elements the n secular equations.
For example,
the first element of the result is
€
H11c1 + H12 c 2 + ! + H1nc n
− E { S11c1 + S12 c 2 + ! + S1nc n} = 0.
(21)
A little rearrangement of this equation gives the form of the first secular equation, Eq. (13) with k=1,
€
( H11 −
€
E S11) c1 + ( H12 − E S12 ) c 2 + ! + ( H1n − E S1n ) c n = 0.
(22)
4
Appendix: Matrices and Matrix Operations
This is intended to be only a brief discussion describing the properties and operations of matrices. For additional
information, see any linear algebra text. Throughout this discussion, matrices are distinguished by the use of bold
Helvetica type; for example, A, H, and R, are all matrices.
Definitions
A matrix is an array of numbers or variables arranged in rows and columns. One of the elements of the matrix A
would be labeled Aij , which refers to the element in the ith row and jth column of A. Matrices may be rectangular
or square. A matrix with only one column is called a vector. A matrix with M rows and N columns is denoted as an
M×N matrix. An example of a 3×3 matrix B is shown below:
"B11 B12 B13 %
$
'
B = $B 21 B 22 B 23 ' .
$#B 31 B 32 B 33 '&
(A1)
Transpose of a Matrix
The transpose of a matrix is a rearrangement
of the elements in a matrix. The transpose of a matrix R, denoted RT,
€
is formed by rearranging the elements of R such that
(R )
T
= R ji .
(A2)
" 1 2 -3 %
$
'
R = $-4 4 2 '
$# 1 -5 10'&
(A3)
" 1 -4 1 %
$
'
= $ 2 4 -5 '
$#-3 2 10'&
(A4)
ij
An example 3×3 matrix R and its transpose are shown below.
€
€
R
T
Inverse of a Matrix
–1
The inverse of a matrix, denoted R for
R
€ the matrix R, is a rearrangement of the elements in a matrix such that
–1
R = I. Here, the matrix I is the unit matrix, which consists of zeroes everywhere except down the diagonal. The
diagonal elements are 1. An example 4×4 unit matrix is shown below.
"1
$
0
I = $
$0
$
#0
€
0
1
0
0
0
0
1
0
0%
'
0'
.
0'
'
1&
(A5)
5
Matrix Multiplication
Matrix multiplication consists of multiplying the elements in two matrices together in a certain manner. The result
of matrix multiplication can be either another matrix or just a number (you can think of a single number as a 1×1
matrix). The product of two matrices AB is defined only if the number of columns of A equals the number of rows
of B. Thus, an M×N matrix can be multiplied by an N×K matrix; the result is an M×K matrix. Matrix multiplication
is not commutative. In general, the product AB is not equal to the product BA.
For the product C = AB, the elements of the matrix C are given by
Cij =
∑ Aik Bkj .
(A6)
k
Thus, for the product of two 2×2 matrices A and B, the result is a 2x2 matrix C,
€
#C11 C12 &
#A11 A12 & #B11 B12 &
C = A × B = %
( = %
(%
(
$C21 C22 '
$A21 A22 ' $B 21 B 22 '
" A B + A12B 21
= $ 11 11
# A21 B11 + A22B 21
€
(A7)
A11B12 + A12B 22 %
' .
A21B12 + A22B 22 &
If A is a 3×4 matrix and B is a 4×1 matrix, the product C will be a 3×1 matrix,
€
C =
# C11 &
# A11 A12
%
(
%
A × B = %C21( = %A21 A22
%$C 31('
%$A 31 A 32
#B11 &
A14 & %
(
( %B 21(
A24 (
%B 31(
A 33 A 34 (' %
(
$B 41'
A13
A23
" A11 B11 + A12B 21 + A13B 31 + A14B 41 %
$
'
= $ A21 B11 + A22B 21 + A23B 31 + A24B 41 ' .
$#A 31 B11 + A 32B 21 + A 33B 31 + A 34B 41'&
€
€
(A8)