Permutations

For 5 electrons in 5 M.O.s there are 252 microstates ( 10!/(5!5!)), but as states of different spin do not mix, we can use a smaller number. If doublet states are needed, then 100 states ( 5!/(2!3!)(5!/3!2!) are needed. If only quartet states are of interest, then 25 states ( 5!/(1!4!)(5!/4!1!) are needed and if the sextet state is required, then only one state is calculated.

In the microstates listed, state 1 is the ground-state configuration. This can be written as (2,2,1,0,0), meaning that M.O.s 1 and 2 are doubly occupied, M.O. 3 is singly occupied by an alpha electron, and M.O.s 4 and 5 are empty. Microstate 1 has a component of spin of 1/2, and is a pure doublet. By Kramer's degeneracy--sometimes called time-inversion symmetry--microstate 2 is also a doublet, and has a spin of 1/2 and a component of spin of -1/2.

Microstate 3, while it has a component of spin of 1/2, is not a doublet, but is in fact a component of a doublet, a quartet and a sextet. The coefficients of these states can be calculated from Wigner's symbol, also called the Clebsch-Gordan 3-J symbol . Thus, the coefficient in the doublet is $sqrt{1/2}$( $j_1=3/2,), in the quartet is $sqrt{4/10}$( $j_1=3/2,), and in the sextet, $sqrt{1/10}$( $j_1=3/2,).

Microstate 4 is a pure sextet. If all 100 microstates of component of spin = 1/2 were used in a C.I., one of the resulting states would have the same energy as the state resulting from microstate 4.

Microstate 5 is an excited doublet, and microstate 6 is an excited state of the system, but not a pure spin-state.

By default, if n M.O.s are included in the MECI, then all possible microstates which give rise to a component of spin = 0 for even electron systems, or 1/2 for odd electron systems, will be used.   

 

Table 1:

 Sets of Microstates for Various MECI Calculations

 

Odd Electron Systems

 

Even Electron Systems

 

 

Alpha Beta

 

No. of
Configs.

Alpha Beta

 

No. of
Configs.

C.I.=1

(1,1)x(0,1)

=

1

(1,1)x(1,1)

=

1

2

(1,2)x(0,2)

=

2

(1,2)x(1,2)

=

4

3

(2,3)x(1,3)

=

9

(2,3)x(2,3)

=

9

4

(2,4)x(1,4)

=

24

(2,4)x(2,4)

=

36

5

(3,5)x(2,5)

=

100

(3,5)x(3,5)

=

100


(n,m) means n electrons in m M.O.s.

MOPAC is configured to allow a maximum of MAXCI states (currently: 20,000), where MAXCI is defined in the file meci_C.F90. If more states are needed (see  Table 1), then MAXCI in meci_C.F90 should be modified. Of course, if MAXCI is changed, MOPAC should be recompiled.

If CIS, CISD, or CISDT are specified, then the number of microstates is defined by C.I.=k and the keyword. The number of microstates is a function of k. Let n and m be integers, such that:

begin{displaymath}n=frac{k}{2}end{displaymath}
begin{displaymath}m=frac{k+1}{2}end{displaymath}

If k is odd, then round down to the next lower integer. Then the number of microstates nCIS, nCISD, and nCISDT, for even-electron systems is:

begin{displaymath}begin{array}{lcll}n_{CIS

Note that when CIS is used, the ground state is not included in the list of microstates. Values for the more important k are given in  Table 2.

 

Table 2:

 Number of Microstates for CIS, CISD, and CISDT

C.I.=k

CIS

CISD

CISDT

1

0

1

1

2

2

4

4

3

4

9

9

4

8

27

35

5

12

55

91

6

18

118

282

7

24

205

635

8

32

361

1545


(for even electron systems only)

Footnotes

3-J symbol
The symbol is of form

<j1j2m1m2|j1j2jm>

=

$displaystyle left {frac{(j+m)!(j-m)!(j_1-m_1)!(j_2-m_2)!(j_1+j_2-j)!(2j+1)}......1+m_1)!(j_2+m_2)!(j_1-j_2+j)!(j_2-j_1+j)!(j_1+j_2+j+1)!}right }^{frac{1}{2}}$

 

 

$displaystyle delta(m,m_1+m_2)sum_r(-1)^{j_1+r-m_1}frac{(j_1+m_1+r)!(j_2+j-r-m_1)!}{r!(j-m-r)!(j_1-m_1-r)!(j_2-j+m_1+r)!}$
where the summation is over all values of r such that all factorials occurring are of non-negative integers (0!=1). See [64]. To use the symbol, the coefficient of momentum (j,m) due to two momenta (j1,m1) and (j2,m2) is <j1j2m1m2|j1j2jm>