Chapter 21. A Simple Example of Using Configuration Interaction

Table of Contents

Input File (ci/ci_1.dat):
Output File (ci/ci_1.out):

AMPAC has a very powerful and robust configuration interaction (CI) engine, so it is worthwhile to present a simple example of its use. (A complete description of CI theory and keywords can be found in Chapter 11, Configuration Interaction.) In this example, ethylene is being optimized with a very limited CI. Many more CI examples can be found in the AMPAC test stuite.

Input File (ci/ci_1.dat):

am1 c.i.=2 cistate=3 cimax=100 singlet t=auto truste lforce bonds    1
D2h Ethylene, AM1/CI,  HOMO LUMO Active, S0 (Singlet Ground State)
Opt+LowFreq, Active MOs and Bonds, Calc 3 Lowest States
 C              0.000000  0    0.000000  0    0.000000  0    0    0    0
 C              1.325916  1    0.000000  0    0.000000  0    1    0    0
 H              1.098266  1  122.715971  1    0.000000  0    1    2    0
 H              1.098266  1  122.715971  1 -180.000000  1    1    2    3
 H              1.098266  1  122.715971  1    0.000000  1    2    1    4
 H              1.098266  1  122.715971  1  180.000000  1    2    1    4
 0              0.000000  0    0.000000  0    0.000000  0    0    0    0
	

1

C.I.=2 keyword requests that 2 orbitals surrounding the HOMO-LUMO gap be treated as active in a CAS-CI. These two orbitals correspond to HOMO and LUMO respectively. CISTATE=3 specifies that the three lowest energy CI states will be determined and printed in the output.

Output File (ci/ci_1.out):

 Timestamp: 2011-08-31-12-35-20-0000001734-win64
 User Info: John Millam, Nahum, 
 *******************************************************************************
                         AM1 CALCULATION RESULTS
 *******************************************************************************
 *                             AMPAC Version 10.0.1
 *                                Presented by:
 *
 *                           Semichem, Inc.
 *                           www.semichem.com
 *
 *  AM1      - THE AM1 HAMILTONIAN TO BE USED
 *  RHF      - RESTRICTED HARTREE-FOCK CALCULATION
 *  TRUSTE   - MINIMIZE ENERGY USING TRUST REGION METHOD
 *  LFORCE   - LOWEST IR FREQUENCIES CALCULATION SPECIFIED
 *  T=AUTO   - AUTOMATIC DETERMINATION OF ALLOWED TIME
 *  C.I.=N   -  2 M.O.S TO BE USED IN C.I.
 *  CIMAX= 100 - ALLOWED SIZE FOR CI MATRIX
 *  CISTATE=  3 - EIGENSTATES CALCULATED IN CI
 *  BONDS    - PRINT NON-ZERO ELEMENTS OF FINAL BOND-ORDER MATRIX
 *  SINGLET  - IS THE REQUIRED SPIN MULTIPLICITY
 *******************************************************************************
 AM1 C.I.=2 CISTATE=3 CIMAX=100 SINGLET T=AUTO TRUSTE LFORCE BONDS
 D2h Ethylene, AM1/CI,  HOMO LUMO Active, S0 (Singlet Ground State)
 Opt+LowFreq, Active MOs and Bonds, Calc 3 Lowest States
    ATOM    CHEMICAL   BOND LENGTH    BOND ANGLE    TWIST ANGLE
   NUMBER   SYMBOL     (ANGSTROMS)     (DEGREES)     (DEGREES)
    (I)                   NA:I          NB:NA:I      NC:NB:NA:I     NA    NB    NC
      1     C 
      2     C          1.32592 *                                     1
      3     H          1.09827 *      122.71597 *                    1     2
      4     H          1.09827 *      122.71597 *  -180.00000 *      1     2     3
      5     H          1.09827 *      122.71597 *     0.00000 *      2     1     4
      6     H          1.09827 *      122.71597 *   180.00000 *      2     1     4

   MOLECULAR POINT GROUP            SYMMETRY CRITERIA
            D2h                         0.10000000

          SINGLET STATE CALCULATION

          RHF CALCULATION, NO. OF DOUBLY OCCUPIED LEVELS =       6

        **  REFERENCES TO PARAMETERS  **

 H  (AM1):  M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985).
 C  (AM1):  M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985).

          CARTESIAN COORDINATES
      ATOM            X               Y               Z
       1 C        0.00000000      0.00000000      0.00000000
       2 C        1.32591600      0.00000000      0.00000000
       3 H       -0.59358517      0.92403726      0.00000000
       4 H       -0.59358517     -0.92403726      0.00000000
       5 H        1.91950117     -0.92403726      0.00000000
       6 H        1.91950117      0.92403726      0.00000000

 STANDARD DEVIATION ON ENERGY   (KCAL)       0.00000055519
 STANDARD DEVIATION ON GRADIENT (KCAL/A,RD,RD)  0.00008233 0.00007642 0.00008301

 LOWEST IR FREQUENCIES CALCULATION (MARCH 1999)
 HEAT OF FORMATION=          8.261408 kcal/mole
 RMS GRADIENT NORM=          0.020640 kcal/mole/A
 HESSIAN SPANNED BY  12 INTERNAL COORDINATES.

  1 LOWEST EIGENVALUES OF THE HESSIAN HAVE BEEN ACCURATELY CALCULATED.
 NON ZERO EIGENVALUES, (STD DEV) AND ASSOCIATED EIGENVECTORS: (Angstroms or radians)
  1.51D+01  0.000  0.000  0.000  0.000  0.000 -0.703  0.000  0.000 -0.599  0.000
(1.61D-03)  0.000 -0.384
 NOTE: WAVE NUMBERS ARE BIASED WITH RESPECT TO EXACT VALUES,
       BUT SIGNS ARE ASCERTAINED (UNLESS A ERROR BAR TOO LARGE).

 VIBRATIONAL FREQUENCIES AND ERRORS (CM-1),
 REDUCED FORCE CONSTANTS (MILLIDYNES/ANGSTROMS),
 DIPOLE DERIVATIVES (DEBYE/ANGSTROMS),
 IR INTENSITIES (DEBYE**2/ANGSTROMS**2),
 AND NORMAL MODES (CARTESIAN COORDINATES):
 FREQ  :    0.000    0.000    0.000    0.000    0.000    0.000  965.219
 ERROR :    0.000    0.000    0.000    0.000    0.000    0.000    0.155
 F-CST :  0.00000  0.00000  0.00000  0.00000  0.00000  0.00000  0.27443
 DIP(X):    0.000    0.000    0.000    0.000    0.000    0.000    0.000
 DIP(Y):    0.000    0.000    0.000    0.000    0.000    0.000    0.000
 DIP(Z):    0.000    0.000    0.000    0.000    0.000    0.000    0.000
 DIP TOT    0.000    0.000    0.000    0.000    0.000    0.000    0.000
 IR ITEN    0.000    0.000    0.000    0.000    0.000    0.000    0.000
   1C (x) -0.0043   0.0000   0.0000   0.0000   0.1888   0.0000   0.0000
   1C (y) -0.1256  -0.0465  -0.1542  -0.0091  -0.0029  -0.1249   0.0000
   1C (z)  0.0442   0.0642  -0.1598  -0.1073   0.0010   0.1367   0.1244
   2C (x) -0.0043   0.0000   0.0000   0.0000   0.1888   0.0000   0.0000
   2C (y)  0.1630  -0.0474  -0.1029   0.0239   0.0037  -0.1321   0.0000
   2C (z)  0.0126   0.1237   0.0401  -0.1893   0.0003  -0.0944  -0.1244
   3H (x) -0.2036   0.0006  -0.0354  -0.0228   0.1842   0.0049   0.0000
   3H (y) -0.2523  -0.0460  -0.1768  -0.0236  -0.0057  -0.1217   0.0000
   3H (z)  0.0402   0.4418  -0.3240   0.2038   0.0009   0.1806  -0.3947
   4H (x)  0.1950  -0.0006   0.0354   0.0228   0.1933  -0.0049   0.0000
   4H (y) -0.2523  -0.0460  -0.1768  -0.0236  -0.0057  -0.1217   0.0000
   4H (z)  0.0761  -0.3655  -0.1709  -0.3463   0.0017   0.2956  -0.3947
   5H (x)  0.1950  -0.0006   0.0354   0.0228   0.1933  -0.0049   0.0000
   5H (y)  0.2896  -0.0478  -0.0804   0.0384   0.0066  -0.1352   0.0000
   5H (z)  0.0166  -0.2538   0.2043  -0.5003   0.0004  -0.1383   0.3948
   6H (x) -0.2036   0.0006  -0.0354  -0.0228   0.1842   0.0049   0.0000
   6H (y)  0.2896  -0.0478  -0.0804   0.0384   0.0066  -0.1352   0.0000
   6H (z) -0.0193   0.5534   0.0512   0.0498  -0.0004  -0.2533   0.3948

 AM1 C.I.=2 CISTATE=3 CIMAX=100 SINGLET T=AUTO TRUSTE LFORCE BONDS
 D2h Ethylene, AM1/CI,  HOMO LUMO Active, S0 (Singlet Ground State)
 Opt+LowFreq, Active MOs and Bonds, Calc 3 Lowest States

     GEOMETRY OPTIMIZED : ENERGY MINIMIZED
     SCF FIELD WAS ACHIEVED

                              AM1 CALCULATION
                                                            VERSION 10.0.1

                                                       Aug-31-2011
 
          FINAL HEAT OF FORMATION   =         8.261408 kcal (CI SINGLET No  1)    1
                                    =        34.573991 kJ
          ELECTRONIC ENERGY         =      -736.450184 eV
          CORE-CORE REPULSION       =       425.733187 eV
          TOTAL ENERGY              =      -310.716996 eV
          GRADIENT NORM             =         0.071497 
          RMS GRADIENT NORM         =         0.020639 
          UNSTABLE MODE(S)          =         0 ( ACCURATE  )
          MOLECULAR WEIGHT          =        28.053600 
          MOLECULAR POINT GROUP     = D2h     0.100000
          NO. OF FILLED LEVELS      =         6 (OCC = 2)
          TOTAL NUMBER OF ORBITALS  =        12
          SCF + CI CALCULATIONS     =        16
          COMPUTATION TIME          =         0.23     SECONDS

    ATOM    CHEMICAL   BOND LENGTH    BOND ANGLE    TWIST ANGLE    2
   NUMBER   SYMBOL     (ANGSTROMS)     (DEGREES)     (DEGREES)
    (I)                   NA:I          NB:NA:I      NC:NB:NA:I     NA    NB    NC
      1     C 
      2     C          1.34092 *                                     1
      3     H          1.09708 *      122.43943 *                    1     2
      4     H          1.09708 *      122.43943 *  -180.00000 *      1     2     3
      5     H          1.09708 *      122.43943 *     0.00000 *      2     1     4
      6     H          1.09708 *      122.43943 *   180.00000 *      2     1     4

   MOLECULAR POINT GROUP            SYMMETRY CRITERIA
            D2h                         0.10000000

          RHF EIGENVALUES
    -32.96035    -21.92659    -15.75830    -14.25910    -11.89512    -10.45383
      1.37797      4.05778      4.39209      5.06914      5.56126      5.74277

          CONFIGURATION INTERACTION CALCULATION    3
          4 MICRO-STATES GENERATED BY   CAS-CI  VS         44 ROOM AVAILABLE.
     4 MICRO-STATES FINALLY KEPT.

 CI-ACTIVE MOLECULAR ORBITALS:

 ROOT NO.          6        7    4
             -10.454    1.378
    1 C  S    0.0000   0.0000
    1 C  Px   0.0000   0.0000
    1 C  Py   0.0000   0.0000
    1 C  Pz   0.7071  -0.7071
 
    2 C  S    0.0000   0.0000
    2 C  Px   0.0000   0.0000
    2 C  Py   0.0000   0.0000
    2 C  Pz   0.7071   0.7071
 
    3 H  S    0.0000   0.0000
 
    4 H  S    0.0000   0.0000
 
    5 H  S    0.0000   0.0000
 
    6 H  S    0.0000   0.0000
 

 DETAILED COUNT OF THE    4 CALCULATED LOWEST EIGENSTATES:    5
        SINGLET TRIPLET
    CSF       3       1
 STATES       3       1
 THE EIGENSTATE SELECTED IS No   1 (SINGLET)

 ROW: MAIN MICRO-STATES OVER THE    4 SELECTED IN C.I.    6
 COLUMN: EIGENSTATES FROM  1 TO   3
 
 MO: 00  1:SINGLET  2:TRIPLET  3:SINGLET
   : 67 eV: 0.0000     2.9456     6.6045
   1 20        96%         0%         0%
         ( 0.9807)  ( 0.0000)  ( 0.0000)
   2 +-         0%        50%        50%
         ( 0.0000)  ( 0.7071)  (-0.7071)
   3 -+         0%        50%        50%
         ( 0.0000)  ( 0.7071)  ( 0.7071)
   4 02         4%         0%         0%
         (-0.1953)  ( 0.0000)  ( 0.0000)

 TRANSITION DIPOLE (A.U.) AND OSC. STRENGTHS FROM STATE  1 (SINGLET) TO OTHERS    7
 STATE  eV       nm       X       Y       Z    STRENGTH
   2   2.946    420.9                                 FORBIDDEN TO TRIPLET
   3   6.605    187.7  1.4073  0.0000  0.0000  0.3205
 SUM OF STRENGTHS:     0.9615  0.0000  0.0000

          NET ATOMIC CHARGES AND DIPOLE CONTRIBUTIONS    8
      ATOM            CHARGE        ATOM ELECTRON DENSITY
       1 C           -0.2193          4.2193
       2 C           -0.2193          4.2193
       3 H            0.1097          0.8903
       4 H            0.1097          0.8903
       5 H            0.1097          0.8903
       6 H            0.1097          0.8903

 DIPOLE (DEBYE)   X         Y         Z       TOTAL
 POINT-CHG.     0.000     0.000     0.000     0.000
 HYBRID         0.000     0.000     0.000     0.000
 SUM            0.000     0.000     0.000     0.000


          CARTESIAN COORDINATES    9
      ATOM            X               Y               Z
       1 C        0.00000000      0.00000000      0.00000000
       2 C        1.34092452      0.00000000      0.00000000
       3 H       -0.58848034      0.92588760      0.00000000
       4 H       -0.58848034     -0.92588760      0.00000000
       5 H        1.92940486     -0.92588760      0.00000000
       6 H        1.92940486      0.92588760      0.00000000


          ATOMIC ORBITAL ELECTRON POPULATIONS    10
      1.24839      0.95422      1.01669      1.00000      1.24839      0.95422
      1.01669      1.00000      0.89035      0.89035      0.89035      0.89035

                    BOND ORDERS AND VALENCIES    11

           |        1 C 
       1 C |     3.928231
 
           |        1 C        2 C 
       2 C |     1.852496   3.928231
 
           |        1 C        2 C        3 H 
       3 H |     0.957752   0.006722   0.987976
 
           |        1 C        2 C        3 H        4 H 
       4 H |     0.957752   0.006722   0.008643   0.987976
 
           |        1 C        2 C        3 H        4 H        5 H 
       5 H |     0.006722   0.957752   0.013244   0.001615   0.987976
 
           |        1 C        2 C        3 H        4 H        5 H        6 H 
       6 H |     0.006722   0.957752   0.001615   0.013244   0.008643   0.987976
 
 
     ELAPSED WALL CLOCK TIME :      0.24 SECONDS
     FULL COMPUTATION TIME :      0.23 SECONDS
	

1

The primary CI eigenstate is, as requested, the ground state singlet, S0. For reference, the AM1/SCF and experimental heats of formation are 16.5 kcal/mol and 12.5 kcal/mol, respectively. There is a significant decrease in energy in going from AM1/SCF to even this minimal AM1/CAS-CI. Since AM1 (and all of the semi-empirical models in AMPAC) was parameterized against experiment at the SCF level, absolute heats of formation at the corresponding CI level are generally too low, especially at higher levels of CI.

2

This Z-matrix shows the AM1 optimized geometry of Ethylene in the primary CI eigenstate. All of the results in AMPAC output file, including those for the secondary CI eigenstates, are calculated at this geometry. The transition energies between the primary and secondary CI eigenstates are thus vertical transition energies.

3

As requested, the number of CI-active MOs was 2, the HOMO and LUMO. There are six possible microstates, given below in terms of the SO occupancies of the HOMO and LUMO, where + means the alpha SO is occupied, - means the beta SO is occupied and 0 means the corresponding SO is unoccupied:

Table 21.1. Possible microstates with 2 CI-active MOs

# HOMO LUMO S z
1 (+-) (00) 0
2 (+0) (0-) 0
3 (0-) (+0) 0
4 (00) (+-) 0
5 (+0) (+0) 1
6 (0-) (0-) -1

This minimal CAS-CI for a singlet state uses the first four 4 microstates, which have Sz = 0. The 5th and 6th microstates, with Sz = 1 and -1, are degenerate with the linear combination of the 3rd and 4th microstates and so are not used, even for the triplet CI eigenstates (they cannot be used for the singlet states, of course).

4

The AO coefficients of the two CI-active MOs show that the first (the HOMO) is a π MO while the second (the LUMO) is a corresponding π* MO.

5

The 3 lowest secondary CI eigenstates were calculated in addition to the primary one S0. One of the secondary CI eigenstates is a triplet, the others being singlets. The CSF row in the table refers to the number of spin-adapted configurations used in the expansion of the CI eigenstates.

6

This table gives the contribution - as both a percentage and as a normalized coefficient - of each microstate to each of the 3 requested CI eigenstates, whose energies are given (in eV) relative to the primary eigenstate. The SO occupancies of the CI-active MOs for each microstate are also given. Thus, the first excited state, 2:TRIPLET (T1), has an energy of -307.7714 (= -310.716996 + 2.9456) eV and it is an equal combination (50% / 50%) of the the second and third microstates. In the second microstate, the MOs 6 (HOMO) and 7 (LUMO) are both singly occupied (+- means that only the alpha SO of the HOMO is occupied while only the beta SO of the LUMO is occupied). The third microstate is the spin‑flipped complement to the second, with the beta SO of the HOMO being occupied and the alpha SO of the LUMO being occupied. The primary CI eigenstate, 1:SINGLET (ground state S0), is nearly identical to the first microstate, which is the SCF reference determinant (20 means 2 electrons in the HOM0 and 0 electrons in the LUMO). The fourth microstate, in which the LUMO is doubly occupied, makes only a 4% contribution to the primary CI eigenstate and no contribution to the others. Note that the Sz eigenvalue is 0 for all microstates.

7

This table gives the transition energies, transition wavelengths, transition dipoles and oscillator strengths between the primary CI eigenstate and the two secondary CI eigenstates. Since the primary CI eigenstate is a singlet, the transition dipole and corresponding oscillator strength for the triplet CI eigenstate is identically zero. The first excited singlet state (S1) has a significant transition dipole parallel to the C-C bond and the corresponding oscillator strength of 0.3025 indicates there should be a significant absorption intensity around 187 nm for gas phase ethylene.

8

The atomic Mulliken charges and dipole moments given in this section are for the primary CI eigenstate. Values for the secondary eigenstates can be calculated with the AMPAC keyword CIDIP.

9

This table gives the atoms’ Cartesian coordinates for the geometry optimized in the primary CI eigenstate.

10 11

These tables of AO electron populations, bond orders and valencies are calculated from the first-order density matrix of the primary CI eigenstate.