7.8 Coupled-Cluster Excited-State and Open-Shell Methods

7.8.24 Interpretation of EOM/CI Wave Functions and Orbital Numbering

Analysis of the leading wave function amplitudes is always necessary for determining the character of the state (e.g., HOMOLUMO excitation, open-shell diradical, etc.). The CCMAN module print out leading EOM/CI amplitudes using its internal orbital numbering scheme, which is printed in the beginning. The typical CCMAN EOM-CCSD output looks like:

Root 1 Conv-d yes Tot Ene= -113.722767530 hartree (Ex Ene 7.9548 eV),
U1^2=0.858795, U2^2=0.141205 ||Res||=4.4E-07
Right U1:
       Value                   i            ->    a
       0.5358                  7( B2  ) B   ->   17( B2  ) B
       0.5358                  7( B2  ) A   ->   17( B2  ) A
      -0.2278                  7( B2  ) B   ->   18( B2  ) B
      -0.2278                  7( B2  ) A   ->   18( B2  ) A

This means that this state is derived by excitation from occupied orbital #7 (which has b2 symmetry) to virtual orbital #17 (which is also of b2 symmetry). The two leading amplitudes correspond to ββ and αα excitation (the spin part is denoted by A or B). The orbital numbering for this job is defined by the following map:

The orbitals are ordered and numbered as follows:
Alpha orbitals:
Number  Energy    Type    Symmetry  ANLMAN number  Total number:
  0    -20.613     AOCC      A1      1A1     1
  1    -11.367     AOCC      A1      2A1     2
  2     -1.324     AOCC      A1      3A1     3
  3     -0.944     AOCC      A1      4A1     4
  4     -0.600     AOCC      A1      5A1     5
  5     -0.720     AOCC      B1      1B1     6
  6     -0.473     AOCC      B1      2B1     7
  7     -0.473     AOCC      B2      1B2     8

  0      0.071     AVIRT     A1      6A1     9
  1      0.100     AVIRT     A1      7A1     10
  2      0.290     AVIRT     A1      8A1     11
  3      0.327     AVIRT     A1      9A1     12
  4      0.367     AVIRT     A1     10A1     13
  5      0.454     AVIRT     A1     11A1     14
  6      0.808     AVIRT     A1     12A1     15
  7      1.196     AVIRT     A1     13A1     16
  8      1.295     AVIRT     A1     14A1     17
  9      1.562     AVIRT     A1     15A1     18
 10      2.003     AVIRT     A1     16A1     19
 11      0.100     AVIRT     B1      3B1     20
 12      0.319     AVIRT     B1      4B1     21
 13      0.395     AVIRT     B1      5B1     22
 14      0.881     AVIRT     B1      6B1     23
 15      1.291     AVIRT     B1      7B1     24
 16      1.550     AVIRT     B1      8B1     25
 17      0.040     AVIRT     B2      2B2     26
 18      0.137     AVIRT     B2      3B2     27
 19      0.330     AVIRT     B2      4B2     28
 20      0.853     AVIRT     B2      5B2     29
 21      1.491     AVIRT     B2      6B2     30

The first column is CCMAN’s internal numbering (e.g., 7 and 17 from the example above). This is followed by the orbital energy, orbital type (frozen, restricted, active, occupied, virtual), and orbital symmetry. Note that the orbitals are blocked by symmetries and then ordered by energy within each symmetry block, (i.e., first all occupied a1, then all a2, etc.), and numbered starting from 0. The occupied and virtual orbitals are numbered separately, and frozen orbitals are excluded from CCMAN numbering. The two last columns give numbering in terms of the final ANLMAN printout (starting from 1), e.g., our occupied orbital #7 will be numbered as 1B2 in the final printout. The last column gives the absolute orbital number (all occupied and all virtuals together, starting from 1), which is often used by external visualization routines.

CCMAN2 numbers orbitals by their energy within each irrep keeping the same numbering for occupied and virtual orbitals. This numbering is exactly the same as in the final printout of the SCF wave function analysis. Orbital energies are printed next to the respective amplitudes. For example, a typical CCMAN2 EOM-CCSD output will look like that:

 EOMEE-CCSD transition 2/A1
 Total energy = -75.87450159 a.u.  Excitation energy = 11.2971 eV.
 R1^2 = 0.9396  R2^2 = 0.0604  Res^2 = 9.51e-08

 Amplitude    Orbitals with energies
  0.6486       1 (B2) A                  ->    2 (B2) A
              -0.5101                          0.1729
  0.6486       1 (B2) B                  ->    2 (B2) B
              -0.5101                          0.1729
 -0.1268       3 (A1) A                  ->    4 (A1) A
              -0.5863                          0.0404
 -0.1268       3 (A1) B                  ->    4 (A1) B
              -0.5863                          0.0404

which means that for this state, the leading EOM amplitude corresponds to the transition from the first b2 orbital (orbital energy -0.5101) to the second b2 orbital (orbital energy 0.1729).

The most complete analysis of EOM-CC calculations is afforded by deploying a general wave-function analysis tool contained in the libwa module and described in Section 11.2.6. The EOM-CC state analysis is activated by setting STATE_ANALYSIS = TRUE. In addition, keywords controlling calculations of state and interstate properties should be set up accordingly.

Note:  Wave function analysis is only available for CCMAN2.

Example 7.89  Wave function analysis of the EOM-IP states (He+3).

$molecule
   0 1
   He
   He   1       R1
   He   2       R1      1       A

   R1 = 1.236447
   A  = 180.00
$end

$rem
   METHOD            = EOM-CCSD
   BASIS             = 6-31G
   IP_STATES         = [1,0,0,0,0,1,0,0]
   CC_EOM_PROP       = true   Analyze state properties (state OPDM)
   CC_STATE_TO_OPT   = [1,1]  Compute transition properties wrt 1st EOM state of 1st irrep
   CC_TRANS_PROP     = true   Analyze transitions (transition OPDM)
   STATE_ANALYSIS    = true
   MOLDEN_FORMAT     = true
   NTO_PAIRS         = 2
$end