As an Amazon Associate, I earn from qualifying purchases. For more practice problems like this, see Schaum’s Outline of College Chemistry, 10th Edition.

1. What the question is asking (in plain language)

A student calculated the electron‑repulsion integrals (ERIs) for a water molecule with a minimal Huzinaga basis set using three quantum‑chemistry programs:

Program How the ERIs are printed
GAMESS ( NPRINT=4 ) every unique integral, without any symmetry factor
Molpro ( INT,SPRI=2 ) unique integrals, but the numbers are sometimes multiplied by 2, 4 or 8
Gaussian 09 ( SCF(Conventional) IOp(3/33=6) Symmetry=None ) the printed numbers look “odd”: some are 2‑ or 4‑times larger, some have opposite sign, and many are accompanied by a small integer (1, 2, 4 or 8).

The student wants to know:

  1. Why Gaussian’s output looks different from the other packages.
  2. How to interpret the numbers that Gaussian prints so that they can be compared directly with GAMESS or Molpro.

The answer must explain the role of permutational symmetry (the “redundancy factors”) and any sign conventions that differ between the codes.

2. Background – the definition of an ERI and its symmetry

For a set of (real) basis functions ({\chi_i}) the chemist’s two‑electron integral is

[ (ij|kl)=\iint \chi_i(\mathbf r_1)\,\chi_j(\mathbf r_1)\, \frac{1}{r_{12}}\, \chi_k(\mathbf r_2)\,\chi_l(\mathbf r_2)\; d\mathbf r_1 d\mathbf r_2 . ]

Because the Coulomb operator (1/r_{12}) is symmetric with respect to exchange of the two electrons, the integral obeys eight symmetry relations:

[ \begin{aligned} (ij|kl)&=(ji|kl)=(ij|lk)=(ji|lk)
&=(kl|ij)=(lk|ij)=(kl|ji)=(lk|ji) . \end{aligned} \tag{1} ]

Consequences:

Situation Number of distinct permutations Redundancy factor that appears in the full 4‑index tensor
(i=j,\;k=l,\;i\neq k) (two pairs equal) 2 (the two distinct pairs) 4
All four indices different 8 8
All four indices the same ((i=j=k=l)) 1 1
One pair equal, the other two different (e.g. (i=j\neq k\neq l)) 4 4
… … …

Many codes store only the “unique” integrals (the ones that satisfy a canonical ordering such as (i\ge j,\;k\ge l) and ((i,j)) precedes ((k,l)) in a lexicographic sense). When the integral is later used to build the Fock matrix, the program multiplies the stored value by the appropriate redundancy factor so that the full 8‑fold symmetry of Eq. (1) is restored.

3. What each program actually prints

Program What is printed How to obtain the conventional ((ij kl))
GAMESS (NPRINT=4) The raw unique integral without any factor. Already the conventional value; no further scaling needed.  
Molpro (INT,SPRI=2) The raw unique integral followed by an integer (1, 2, 4, 8) that tells how many symmetry‑equivalent copies exist. The integer is not multiplied into the printed number. Multiply the printed number by the integer to get the value that would appear in the fully symmetrised tensor.  
Gaussian 09 (IOp(3/33)=6) For each integral a line such as     
   ERI   1   2   3   4   =   0.123456   *   4
where the number after the asterisk is the redundancy factor. The number before the asterisk is the unique integral. Multiply the number before the asterisk by the factor after the asterisk. The product is the conventional ((ij kl)) that would be used in a Fock‑matrix construction.

Thus the “odd” numbers that the student saw in Gaussian are simply the unscaled integrals; the factor printed on the same line tells exactly how to rescale them.

3.1 Example from the water calculation

Suppose Gaussian prints

   ERI   2   5   6   7   =  -0.001234567   *   4

Interpretation

  • Basis‑function indices (Cartesian order) → (2,5 6,7) is a unique integral because the indices obey the canonical ordering.
  • The raw value is (-1.234567\times10^{-3}) a.u.
  • The factor 4 tells us that there are four symmetry‑equivalent permutations (e.g. (5 2 6 7), (2 5 7 6), …).

The conventional chemist’s ERI is therefore

[ (25|67)=4\times(-0.001234567) = -0.004938268\;\text{a.u.} ]

If GAMESS printed the same integral, it would already show (-0.004938268).

4. Why some Gaussian numbers appear with the opposite sign

Two additional, subtler, sources of apparent sign differences exist:

  1. Cartesian vs. spherical d (or f) functions
    The Huzinaga minimal basis for oxygen contains a d‑type set (the 2d functions). Gaussian stores them in pure spherical form, whereas many older codes (including GAMESS when run with the default SCF=CONV) keep the Cartesian set ({x^2,\,y^2,\,z^2,\,xy,\,xz,\,yz}). The linear transformation from Cartesian to spherical contains coefficients of alternating sign (e.g. (d_{z^2}= \frac{1}{\sqrt{3}}(2z^2-x^2-y^2))). Consequently, an integral that involves a spherical d function can appear with the opposite sign compared with a Cartesian‑only program.

  2. Ordering of the Cartesian components
    Different programs adopt different internal orderings for the six Cartesian d functions. Gaussian’s order is

    [ xx,\; yy,\; zz,\; xy,\; xz,\; yz, ]

    whereas GAMESS (and the original Huzinaga file) often uses

    [ xx,\; xy,\; xz,\; yy,\; yz,\; zz . ]

    If the same numerical integral is written with a different index ordering, the sign may flip because the transformation matrix contains negative elements (e.g. the conversion from the (zz) component to the spherical (d_{z^2}) has a factor (-\tfrac{1}{2})).

Bottom line: after correcting for the redundancy factor and making sure that the same type (Cartesian vs. spherical) and the same ordering of the d‑functions are used, the numbers from Gaussian agree with GAMESS and Molpro to the printed precision.

5. Step‑by‑step recipe to compare Gaussian’s ERIs with another program

  1. Identify the basis‑function ordering used by each code.
    For the Huzinaga minimal set the typical ordering (Cartesian) is

    1  O 1s
2  O 2s
3  O 2px
4  O 2py
5  O 2pz
6  H1 1s
7  H2 1s

    If the code uses spherical d functions, replace indices 3–5 with the five spherical d’s (the program will tell you the order in the header of the integral printout).

  2. Read each Gaussian line of the form

    ERI   i   j   k   l   =   value   *   factor
    • Compute value_scaled = value * factor.
    • Keep track of the canonical ordering (Gaussian already enforces it).

  3. If comparing to a program that prints Cartesian d integrals, apply the Cartesian‑↔‑spherical transformation to the Gaussian numbers before scaling. The transformation matrix (C) for d‑functions is (Cartesian → spherical):

    [ \begin{pmatrix} d_{xy}\ d_{xz}\ d_{yz}\ d_{x^2-y^2}\ d_{z^2} \end{pmatrix} = C\, \begin{pmatrix} xy\ xz\ yz\ xx-yy\ 2zz-xx-yy \end{pmatrix}, \qquad C = \begin{pmatrix} 1 & 0 & 0 & 0 & 0
    0 & 1 & 0 & 0 & 0
    0 & 0 & 1 & 0 & 0
    0 & 0 & 0 & 1 & 0
    0 & 0 & 0 & 0 & 1 \end{pmatrix} ]

    (In practice you only need the sign of the relevant rows; many libraries already provide a routine to convert integrals.)

  4. Compare the scaled (and, if needed, transformed) numbers with the values printed by GAMESS or Molpro. They should match to the printed number of decimal places (typically 8–10 for a minimal basis).

  5. If a discrepancy remains, double‑check:

    • the integral screening thresholds (Gaussian may discard integrals smaller than (10^{-12}) a.u.; GAMESS may keep them).
    • the convention for the ERI: some codes use the chemist’s ((ij kl)) while others use the physicist’s (\langle ij kl\rangle = (ij kl)). For real basis functions the two are identical, but a sign change can appear for complex spherical harmonics (not an issue for the Huzinaga set).

6. Final answer – why Gaussian looks different and how to read it

*Gaussian prints the raw unique two‑electron integrals and, on the same line, the redundancy factor (1, 2, 4, 8

Original question: Understanding two-electron integrals in Gaussian 09 on Chemistry Stack Exchange, licensed CC BY-SA.