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-Gordon 3-J symbol . Thus, the coefficient in the doublet is

(

), in the quartet is

(

), and in the sextet,

(

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, where MAXCI is defined in the file meci.h. If more states are needed (see Table 1), then MAXCI in meci.h 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:

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

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)

<j₁j₂m₁m₂\|j₁j₂jm>	=	$displaystyle left {frac{(j+m)!(j-m)!(j_1-m_1)!(j_2-m_2)!(j_1+j_2-j)!(2j+1)}......1+m_1)!(j_2+m_2)!(j_1-j_2+j)!(j_2-j_1+j)!(j_1+j_2+j+1)!}right }^{frac{1}{2}}$
		$displaystyle delta(m,m_1+m_2)sum_r(-1)^{j_1+r-m_1}frac{(j_1+m_1+r)!(j_2+j-r-m_1)!}{r!(j-m-r)!(j_1-m_1-r)!(j_2-j+m_1+r)!}$

Permutations

Footnotes