What do the parameters at the top of an FCIDUMP file mean?

Question

A while ago I wrote a simple program that takes in an FCIDUMP file (generated by MOLRPO) and can determine matrix elements (or construct the whole FCI matrix), with the help of this forum in this thread.

However, I didn't use any of the values at the top of the file, except for NORB and NELEC (which in the string representation were just half the length of the string and the number of ones). Presumably these serve to reduce the size of the matrix somehow (maybe something with group theory), or have something to do with the irrep of the determinants. What do these parameters mean?

Perhaps it would also be helpful to work with an example, like the minimal STO-3G basis H2 (separation 1.4 bohr), if I use my program (which is hopefully correct...) I get an FCI matrix:

[[-1.83100004  0.          0.          0.          0.          0.18125791]
 [ 0.         -1.24609329  0.          0.          0.          0.        ]
 [ 0.          0.         -1.06483537  0.18125791  0.          0.        ]
 [ 0.          0.          0.18125791 -1.06483537  0.          0.        ]
 [ 0.          0.          0.          0.         -1.24609329  0.        ]
 [ 0.18125791  0.          0.          0.          0.         -0.25370925]]

However, I know from Szabo and Ostlund (e.g. using pg 162) that this is reduced to

[[-1.8310000369984170       0.18125779940066075]
 [0.18125779940066075      -0.25371106084836703]]

How do I get here given the following?

NORB=  2,NELEC=  2,MS2= 0,
ORBSYM=1,5,
ISYM=1,
OCC=  1,  0,  0,  0,  0,  0,  0,  0,
CLOSED=  1,  0,  0,  0,  0,  0,  0,  0,

And do the parameters give any extra information besides what is in the matrix?

Answer 1

MS2 is related to the spin. Specifically, it is the number of unpaired electrons. Your molecule is a singlet, which is why it says 0.

ORBSYM is the list of symmetries for each orbital. In this case $\ce{H2}$ is in $D_{\infty h}$ but almost all electronic structure codes do not support non-Abelian groups, so instead we would almost always use $D_{2h}$. The FCIDUMP format was invented by Peter Knowles, and the writing and reading of FCIDUMP files would have been implemented in MOLPRO long before other software, so assuming this FCIDUMP was written by MOLPRO, you can figure out what the 1 and 5 mean from the $D_{2h}$ table here (Table 2). This tells us that the first molecular orbital is of type $A_g$ and the second is of type $B_{1u}$. That makes sense because in your minimal basis set with only 2 spatial orbitals, you have $\sigma$ and $\sigma^*$ bonding and anti-bonding orbitals. You mentioned PySCF in the other question, and unfortunately they have a different convention for labeling the irreps of $D_{2h}$, in which $A_g=0$ and $B_{1u}=5$, which is not what you have in your FCIDUMP, perhaps because you didn't use PySCF to print the FCIDUMP, or because the PySCF documentation has an error. My open source "Quantum Chemistry Cheat Sheet" will help you translate between the languages of many of the most popular electronic structure programs, and here I'll just copy and paste from there the relevant information for $D_{2h}$ since you're dealing with $\ce{H2}$:

$D_{2h}$	irrep	MOLPRO	MRCC	CFOUR	PySCF
$s$	$A_g$	1	1	1	0
$p_z$	$B_{1u}$	5	6	5	5
$p_y$	$B_{2u}$	2	7	2	6
$p_x$	$B_{3u}$	3	8	3	7
$d_{xy}$	$B_{1g}$	4	2	4	1
$d_{xz}$	$B_{2g}$	6	3	7	2
$d_{yz}$	$B_{3g}$	7	4	6	3
$d_z^2$	$A_g$	1	1	1	0
$d_{x^2-y^2}$	$A_g$	1	1	1	0
$f_{xyz}$	$A_u$	8	5	8	4

By the way if you don't like how the $d_{x^2 - y^2}$ orbital turns out in the table, you might consider supporting my feature request to fix that bug 😊.

ISYM is the overall symmetry of the molecule. $A_g \times B_{1u} = A_g$, which was labeled as 1 in the program that you used to print that FCIDUMP.

OCC are the occupied orbitals for each irrep of the point group, which in this case is $D_{2h}$ so there's 8 entries there. Only an $A_g$ orbital is occupied, so the entries for all other irreps is 0, and the $A_g$ irrep has only one occupied orbital, which is the $\sigma$ bonding orbital, so we have a 1 there. This means the anti-bonding $\sigma^*$ orbital does not have any electrons in it.

CLOSED are the closed orbitals, with the same convention as OCC. Since there's 2 electrons in the one occupied orbital, we have a "closed" orbital. It would be open if it didn't have 2 electrons in it.