Each MECI calculation invoked by use of the keywords C.I.=n and C.I.=(n,m) normally gives rise to states of quantized spins. When C.I. is used without any other modifying keywords, the number of states shown in the Tables 1 and 2 will be obtained. These numbers of spin states will be obtained irrespective of the chemical nature of the system.
Table 1: States arising from C.I.=n 

No. of M.O.s 
Number of States Arising from 

in MECI 
Odd Electron Systems 
Even Electron Systems 

Doublets 
Quartets 
Sextets 
Singlets 
Triplets 
Quintets 

1 
1 


1 


2 
2 


3 
1 

3 
8 
1 

6 
3 

4 
20 
4 

20 
15 
1 
5 
75 
24 
1 
50 
45 
5 
Table 2: Number of states arising from C.I.=(n,m) 

Number of States Arising from  
Keywords 
Oddelectron systems  Evenelectron systems 
C.I.=(2,1) 
2 = (2!/(0!2!))*(2!/(1!1!)) 
4 
C.I.=(3,1) 
9  9 
C.I.=(3,2)  3  9 
C.I.=(4,1)  24  16 
C.I.=(5,1)  50  25 
C.I.=(8,1) 
224 = (8!/(6!2!))*(8!/(7!1!)) 
64 
C.I.=(8,4) 
3920 = (8!/(3!5!))*(8!/(4!4!)) 
4900 
If the C.I. is not performed correctly then the symmetry features of the system will not be correctly identified. Two systems, methane and a transition metal atom are used to verify that the correct symmetry features are produced.
The following dataset is provided to allow the quantities in this discussion to be reproduced::
vectors c.i.=8 meci Methane H 0.00000000 +0 0.0000000 +0 0.0000000 +0 0 0 0 C 1.08523001 +0 0.0000000 +0 0.0000000 +0 1 0 0 H 1.08523001 +0 109.4712210 +0 0.0000000 +0 2 1 0 H 1.08523001 +0 109.4712210 +0 120.0000000 +0 2 1 3 H 1.08523001 +0 109.4712210 +0 120.0000000 +0 2 1 3
Alternatively, all the data described here can be downloaded.
Methane provides a good test of the configuration interaction calculation. If the C.I. is not performed correctly, the symmetry features of the system will not be correctly identified.
Using the standard semiempirical basis set, there are eight atomic orbitals, 2s, 2px, 2py, and 2pz on carbon, and a 1s on each of the four hydrogen atoms. In the molecule these form eight molecular orbitals that have symmetries, in the order of increasing energy: 1a1, 1t2, 2t2, and 2a1. Methane is a small molecule, only five atoms and 8 molecular orbitals, so evaluation of the various terms in the C.I. calculation is very fast. What is not fast is calculating the states formed when C.I.=8 or C.I.=(8,4) is used.
Both configuration interaction calculations are the same  all eight M.O.s and all eight electrons are used. So the active space consists of eight M.O.s, and permutation of the eight electrons, four of α and four of β spin, in these M.O.s result in (8!*(4!)^{2})^{2} = 4900 microstates or configurations. Diagonalization of the resulting configuration interaction matrix is slow because of the computational effort involved. By default, the spinstates and pointgroup symmetry representations of the resulting states are evaluated; these are shown in the following Table:
Table 3 

Number of States for each representation for Methane 

Spinstate 
Spin 
A1  A2  E  T1  T2 
# States (M_{s} = 0) 
# States (all M_{s}) 

SINGLET  0  104  60  152  196  236  1764  1764  
TRIPLET  1  97  97  191  296  296  2352  7056  
QUINTET  2  31  31  65  88  88  720  3600  
SEPTET  3  7  0  4  4  12  63  441  
NONET  4  1  0  0  0  0  1  9  
Totals:  240  188  412  584  632  4900  12870 
Each state has a set of quantum numbers. In this system, as in all systems that have noninfinite pointgroups, these are given the symbols of the type "nSRM_{s}." In other types of systems, such as the infinite groups, space groups, and magnetic groups, other symbols are used, but these are not relevant here and can be ignored. The quantum number n is 1 for the lowestenergy state of symmetry SRM_{s}, 2 for the next one, and so on. S is the spinstate of the state; this can be 0, 1, 2, 3, or 4 here. Each spinstate has (2S + 1) magnetic components, all of the same energy, so in chemistry these are commonly referred to as Singlet, Triplet, Quintet, Septet, and Nonet. Note: by convention, quantum numbers for atomic and molecular orbitals use lowercase letters, quantum numbers for entire systems, such as a wavefunction for methane, use uppercase letters. That convention will be used here. When discussing individual states, the standard convention is to specify the spinstate in terms of the degeneracy of the state, i.e., the number of magnetic components, so the nonet state A1 would be written as ^{9}A1, and for a specific component of spin, as the subscript suffix, e.g. ^{9}A1_{4}. The quantum number M_{s} refers to the magnetic component of spin, and is most simply defined as half the difference in the number of α and β electrons in each state. Because the number of α and β electrons are the same in all microstates used here, all states have a M_{s} of zero, i.e., M_{s} = 0. From here on, for convenience the M_{s} quantum number will be dropped unless needed. Finally R is the irreducible representation of the pointgroup symmetry of the state. A1 and A2 are nondegenerate, that is, they are used for states that have exactly one component, E is a degenerate state having two components, and T1 and T2 are degenerate states that have three components. For convenience, the degeneracy of a specific irreducible representation can be written D_{R}.
All states are orthogonal to all other states. This orthogonality can be intrinsic, thus all spinstates are automatically orthogonal, or result from diagonalization, thus all states of A1 symmetry are expressed by orthonormal eigenfunctions.
Given that there are 4900 states in this system, the addition of the individual states must add to 4900. This addition is made complicated by the degeneracy of the various individual states. Because of this, the simple addition 104 + 60 + etc. + 97 + 97 + 191 + etc. is replaced by the sum of the product of the degeneracy of each state times the frequency, F, of occurrence of that state. So the simple sum is replaced by (D_{R})F, thus the number of Singlet states is given by: 1×104 + 1×60 + 2×152 + 3×196 + 3×236 = 1764, and the number of Triplet states is 1×97 + 1×97 + 2×191 + 3×296 + 3×296 = 2352. The sum of all these states is 4900, as expected.
This calculation was simplified by only using M_{s} = 0. If that
constraint is removed, so that eight electrons are permuted among 16 molecular
orbitals, i.e., the number of α and β molecular
orbitals, then the number of states would rise to 16!*(8!)^{2} = 12870.
Calculating all these states in one job would be very difficult to perform, as all
12870 microstates would need to be supplied by keyword MICROS, and
diagonalization of such a large matrix would take a long time,
as would the symmetry analysis. Fortunately, states of different M_{s}
are automatically orthogonal, so the set of 12870 microstates can be separated
into nine sets where all microstates in a set have the same M_{s}.
These sets could be represented by M_{s} = 4, 3, 2, 1, 0, +1,
+2, +3, and +4. Because of timereversal symmetry the states resulting from M_{s}
= n are equivalent to those for M_{s} = +n; this reduces the
number of sets from nine to five, vis: M_{s} = 0, 1,
2, 3, and 4. The set with M_{s} = 0 has already
been run, so all that remains is to run the sets M_{s} = 1,
2 3, and 4. Editing the dataset for methane is straightforward; to
select all states that have M_{s} = 1 use the
keyword
As expected, the set of states for M_{s} = 4 consists of only one state, the nonet ^{9}A1, there being only one way to put eight α electrons into eight molecular orbitals.
For the set of states with M_{s} = 3 there are eight ways to put seven α electrons in eight molecular orbitals and eight ways to put one β electron in eight molecular orbitals; this results in eight times eight or 64 microstates and therefore 64 states. The number and symmetries of these states are given by the sum of the entries in SEPTET and NONET in Table 3. All the states are SEPTET except one of the A1 states which is ^{9}A1_{3}; this particular state is part of the ninefold degenerate manifold ^{9}A1_{4:4. }Similarly, the sets of states in M_{s} = 2 and M_{s} = 1 can be understood as the TRIPLET and SINGLET states in Table 3 plus the entries in the next lower line. That many states are degenerate can be verified by examining the associated energy level for the various values of M_{s } in a state, they should all be identical. Movement within a state from one M_{s } to another can be accomplished by using the step operator, otherwise known as the shift or ladder operator, but caution is advised when working with these operators  before starting, some practice sessions are recommended, for example showing the Ŝ_{+}β> = α and that Ŝ_{}αα> = 2^{½}(αβ +βα).
This description can be concluded by explaining the entries in the column "# States." Starting at the bottom, with the number of states for the spinstate NONET, 1, the number of states is given by (8!/(8!*0!))^{2} = 1. On the next line up, the SEPTET line, the number of states is (8!/(7!*1!))^{2}  (8!/(8!*0!))^{2} = 63, and, to generalize, the number of entries is given by (8!/((8n)!*n!))^{2}  (8!/((8(n1))!*(n1)!))^{2} , where "n" is the spin, i.e. ½( number of α electrons  number of β electrons). The first of these two terms is obviously the component of spin, M_{s}, the second can be understood as excluding all states that have a higher total spin.
For users who want to explore the symmetry relationships within pointgroup Td a bit further, the following multiplication table might be useful.
A1 x X = X
E × A2 = E
T1 × A2 = T2
T2 × A2 = T1
E × E = A1 + A2 + E
E × T1 = T1 + T2 = E × T2
T1 × T2 = A2 + E + T1 + T2
T1 × T1 = A1 + E + T1 + T2 = T2 × T2
Transition metal atom states are an amazingly rich field for symmetry analysis. Unlike the quantum numbers for electrons that are very limited, state quantum numbers span a wide range and provide a wealth of detailed information on the atomic structure. A chromium atom is used in the following discussion. This element was chosen as the representative for all transition metals; if you'd prefer to use a different element, feel free to do so, but be aware that all the quantum numbers would need to be recalculated.
Transition metal atoms use nine atomic orbitals, in order of increasing energy these are a set of five dorbitals, one sorbital, and three porbitals. In the interest of generality, all nine atomic orbitals will be used in the following discussions. This provides a fairly complete set of systems and allows some of the unusual features of state functions to be examined. For other work the five dorbitals could be used on their own, in which case the analysis would be much simpler. In order for the symmetry properties of states to be explored, all the math involved in constructing the configuration interaction matrix and all the analysis of the symmetries of the states must be correct. Put another way, an error in either the math of the C.I. or in the analysis of the state eigenfunctions would prevent the symmetry from being identified. Put still another way, the symmetry analysis provides strong evidence that the underlying atomic theory is correct.
Chemists are very familiar with the shapes of atomic orbitals and there is a strong temptation for them to assign physical meaning to these shapes. Physicists, on the other hand, are more comfortable using complex atomic orbitals that lend themselves to being labeled with the magnetic quantum number m_{s}. So for example the five dorbitals are assigned the magnetic quantum numbers of +2, +1, 0, 1, and 2. Both descriptions are valid, and in MOPAC the underlying theory used in generating the states uses chemist's ideas of atomic orbitals, but the symmetry analysis can most easily be done using the physicist's model of quantized magnetic moments. So for convenience, the physicist's model will be used from here on; this allows the nine atomic orbitals to be assigned the following magnetic quantum numbers:
Magnetic Quantum Numbers of Atomic Orbitals  
s  p  p  p  d  d  d  d  d 
0  +1  0  1  +2  +1  0  1  2 
In addition to the magnetic quantum number, each electron has a spin quantum number of +½ or ½, these correspond to the chemist's model of of α and β electrons. Each atomic orbital can hold zero, one or two electrons.
With this information, the process of populating the atomic orbitals can be started. A given configuration of electrons in a set of atomic orbitals is called a microstate. If there are no electrons present, there is only one possible microstate  the empty state. If one electron is present, it can be in any one of the nine atomic orbitals, so there are nine microstates. For two electrons, one of spin +½ and one of spin ½, there are 81 microstates. If both electrons have the same spin, say +½, then this number drops to 36. Because electron spins add, the component of spin momentum of these microstates is +1. A second set of 36 microstates exists when the spin of both electrons is ½ and the total spin momentum is 1. So for two electrons in nine atomic orbitals, the total number of microstates possible is 81 + 36 + 36 = 153.
The following table shows the number of states for all possible numbers of electrons, and the formulae for calculating these numbers. In the table "No. of States" is the number of microstates with the same component of spin momentum. "Total no. of States" is the addition of the individual sets of states with the same component of orbital momentum. Sets of microstates with a negative component of orbital momentum are not shown because they have the same values as the sets with a positive Ms, but these should be included in the sum.
No. Electrons  Ms  Formula  No. of States  Ms  Formula  No. of States  Ms  Formula  No. of States  Ms  Formula  No. of States  Ms  Formula  No. of States  Total no. of States  No. Electrons  
0  0  (9!/(0!9!))^{2}  1  1  0  
1  ½  (9!/(1!8!))×(9!/(0!9!))  9  18  1  
2  0  (9!/(1!8!))^{2}  81  1  (9!/(2!7!))×(9!/(0!9!))  36  153  2  
3  ½  (9!/(2!7!))×(9!/(1!8!))  324  1½  (9!/(3!6!))×(9!/(0!9!))  84  816  3  
4  0  (9!/(2!7!))^{2}  1296  1  (9!/(3!6!))×(9!/(1!8!))  756  2  (9!/(4!5!))×(9!/(0!9!))  126  3060  4  
5  ½  (9!/(3!6!))×(9!/(2!7!))  3024  1½  (9!/(4!5!))×(9!/(1!8!))  1134  2½  (9!/(5!4!))×(9!/(0!9!))  126  8568  5  
6  0  (9!/(3!6!))^{2}  7056  1  (9!/(4!5!))×(9!/(2!7!))  4536  2  (9!/(5!4!))×(9!/(1!8!))  1134  3  (9!/(6!3!))×(9!/(0!9!))  84  18564  6  
7  ½  (9!/(4!5!))×(9!/(3!6!))  10584  1½  (9!/(5!4!))×(9!/(2!7!))  4536  2½  (9!/(6!3!))×(9!/(1!8!))  756  3½  (9!/(7!2!))×(9!/(0!9!))  36  31824  7  
8  0  (9!/(4!5!))^{2}  15876  1  (9!/(5!4!))×(9!/(3!6!))  10584  2  (9!/(6!3!))×(9!/(2!7!))  3024  3  (9!/(7!2!))×(9!/(1!8!))  324  4  (9!/(8!1!))×(9!/(0!9!))  9  43758  8  
9  ½  (9!/(5!4!))×(9!/(4!5!))  15876  1½  (9!/(6!3!))×(9!/(3!6!))  7056  2½  (9!/(7!2!))×(9!/(2!7!))  1296  3½  (9!/(8!1!))×(9!/(1!8!))  81  4½  (9!/(9!0!))×(9!/(0!9!))  1  48620  9  
10  0  (9!/(5!4!))^{2}  15876  1  (9!/(6!3!))×(9!/(4!5!))  10584  2  (9!/(7!2!))×(9!/(3!6!))  3024  3  (9!/(8!1!))×(9!/(2!7!))  324  4  (9!/(9!0!))×(9!/(1!8!))  9  43758  10  
11  ½  (9!/(6!3!))×(9!/(5!4!))  10584  1½  (9!/(7!2!))×(9!/(4!5!))  4536  2½  (9!/(8!1!))×(9!/(3!6!))  756  3½  (9!/(9!0!))×(9!/(2!7!))  36  31824  11  
12  0  (9!/(6!3!))^{2}  7056  1  (9!/(7!2!))×(9!/(5!4!))  4536  2  (9!/(8!1!))×(9!/(4!5!))  1134  3  (9!/(9!0!))×(9!/(3!6!))  84  18564  12  
13  ½  (9!/(7!2!))×(9!/(6!3!))  3024  1½  (9!/(8!1!))×(9!/(5!4!))  1134  2½  (9!/(9!0!))×(9!/(4!5!))  126  8568  13  
14  0  (9!/(7!2!))^{2}  1296  1  (9!/(8!1!))×(9!/(6!3!))  756  2  (9!/(9!0!))×(9!/(5!4!))  126  3060  14  
15  ½  (9!/(8!1!))×(9!/(7!2!))  324  1½  (9!/(9!0!))×(9!/(6!3!))  84  816  15  
16  0  (9!/(8!1!))^{2}  81  1  (9!/(9!0!))×(9!/(7!2!))  36  153  16  
17  ½  (9!/(9!0!))×(9!/(8!1!))  9  18  17  
18  0  (9!/(9!0!))^{2}  1  1  18 
If, instead of using nine atomic orbitals where each orbital can hold up to two electrons, 18 atomic orbitals are used where the first nine are of α spin and the second nine are of β spin, then the total number of states can be calculated more rapidly using the expression 18!/(n!(18n)!) where "n" is the number of electrons. This symmetry relationship allows the calculation of the total number of states to be evaluated in two different ways.
Three microstates are unique in that they do not interact with any other microstates; these are the zero and 18 electron configurations, and the nineelectron configuration when the magnetic component of spin is 4½. Only the nineelectron microstate is interesting, one of the other two is empty, the other is full. In the nineelectron system all the electrons are spinup, so the total spin is 4½, and the resulting unique state has a spindegeneracy of 10, hence it is a Decet.
In the configuration interaction calculation each microstate (configuration) interacts with all the microstates in the set. Solving the sets where the component of spin is 0 or ½ is straightforward when the number of electrons is in the range 0  5 or 13  18. When the number of electrons is in the range 6  12 the computational effort to generate and manipulate the state wavefunctions rises rapidly, and for the nineelectron system several CPU days of effort are required. Any attempt to solve the state wavefunctions using the 18 atomic orbital sets should be regarded as a wasted effort. First, the number of permutations rises rapidly as shown in the "Total no. of States" column, so that the nineelectron system would likely require a few CPU months to run. Second, because states of different component of spin do not interact, verification that the states that differ only by component of spin are degenerate is straightforward. Third, the total spin of a state is readily calculable, and can be used in constructing the state symbol. Finally, the sets of states for 0, 1, and 2 electrons for the various Ms shown in the table can easily be shown to be equivalent to the sets of states for the 18 atomic orbital permutations. If, however, these points do not dissuade, it is possible to model 18 atomic orbital sets by using keyword MICROS.
The following tables show the number of states in terms of their orbital symmetry and their spin. In the column headed "(2L + 1)" are the orbital degeneracies of the terms, and the various entries in the row headed "M = (2S + 1):" gives the spin degeneracies of the terms. Orbital angular momentum is indicated by the letters S, P, D, F, G, H, I, K, and L; these map onto the quantum numbers L = 0, 1, 2, 3, 4, 5, 6, 7, and 8. Lowercase letters "g" and "u" indicate whether the state is symmetric (gerade) with respect to inversion or antisymmetric (ungerade). If a state function is ungerade, it changes sign on inversion, i.e., when the Cartesian coordinates x, y, and z are inverted to give x, y, and z. Column "No. of Terms" gives the sum of the product of the orbital term and its degeneracy. So in the twoelectron system there are three singlet and one triplet D(g) states, so there would be (3 + 1)*5 = 20 terms. This should be equal to the number of states in the column headed "No. of Terms" in the table. In fact, it is correct for almost all, over 99%, of the sets of electrons.













Notes: The totals for the number of terms and terms time spin degeneracy for the 7, 8, and 9 electron systems are incorrect. This was not due to a fault in the configuration interaction calculation or to a fault in the symmetry analysis, instead it was due to perfect degeneracy of various states. Degeneracy of this type is a consequence of the limited C.I., i.e., the use of one "s", three "p", and five "d" atomic orbitals, and because the states were degenerate, the state eigenvectors could not be resolved into their component states. If spacegroup symmetry is not used, by adding keyword NOSYM, then the total numbers of terms are correct, and the perfect degeneracy can be seen in the eigenvalues. Interesting states:

All the material described here can be downloaded.