Examples of INDO calculations

(Examples provided by Prof. Rebecca L. M. Gieseking, Brandeis University)

(1) Basics of INDO/CI Excited States
 

This job illustrates how to compute CIS excited states for benzene using the INDO/S Hamiltonian. Since the INDO/S parameters are intended solely for excited-state properties, the geometries must be obtained at another level of theory.

 

The input file is:

 

INDO CIS

benzene

 

H           0.000000 0      2.484212 0      0.000000 0

H           0.000000 0     -2.484212 0      0.000000 0

H           2.151390 0      1.242106 0      0.000000 0

H          -2.151390 0     -1.242106 0      0.000000 0

H          -2.151390 0      1.242106 0      0.000000 0

H           2.151390 0     -1.242106 0      0.000000 0

C           0.000000 0      1.396792 0      0.000000 0

C           0.000000 0     -1.396792 0      0.000000 0

C           1.209657 0      0.698396 0      0.000000 0

C          -1.209657 0     -0.698396 0      0.000000 0

C          -1.209657 0      0.698396 0      0.000000 0

C           1.209657 0     -0.698396 0      0.000000 0

 

By default, the CIS keyword requests that all single excitations within the INDO/S basis are included in the CI matrix. For benzene, there are 15 occupied and 15 virtual orbitals, for a total of 225 single excitations. Adding in the SCF ground state gives a total of 226 electron configurations.

 

The energy of the ground state (state 1) is defined as 0, and the output file contains a list of excited-state energies (states 2-226). The first few excited states are:

 

  CI trans.  energy frequency wavelength oscillator ---------polarization---------    dipole   ------components-----

 st.  symm.    eV      cm-1       nm      strength       x          y         z       moment     x       y       z

 

   2      4.7082902    37975.    263.33  0.000000                                   0.000000   0.000  -0.000  -0.000

   3      5.3612330    43241.    231.26  0.000000                                   0.000000   0.000  -0.000  -0.000

   4      6.2211294    50177.    199.29  0.872452  -0.000049   1.000000  -0.000000  0.000000   0.000  -0.000  -0.000

   5      6.2211297    50177.    199.29  0.872452   1.000000   0.000049  -0.000000  0.000000   0.000  -0.000   0.000

 

The first two excited states (states 2 and 3), with energies of 4.708 and 5.361 eV, are dark states with oscillator strengths of exactly zero. States 4 and 5 are degenerate at an energy of 6.221 eV, and are bright states. State 4 has its transition dipole moment aligned along the y axis, and state 5 has its transition dipole moment aligned along the x axis (see the polarization block within the list of excited states).

 

(2) Solvatochromism

 

This job illustrates how to compute the solvatochromic shift of the first excited state of carbon monoxide between ethanol and toluene using CIS with the INDO/S Hamiltonian, defined as:

 

(Excited-state energy in ethanol) (Excited-state energy in toluene)

 

Since the INDO/S parameters are intended solely for excited-state properties, the geometries must be obtained at another level of theory.

 

The input file in ethanol is:

 

INDO CIS EPS=24.3 N**2=1.85

Carbon monoxide in ethanol

 

C  0.000000 0   0.000000 0   0.000000 0

O  1.173000 0   0.000000 0   0.000000 0

 

The input file in toluene is:

 

INDO CIS EPS=2.38 N**2=1.99

Carbon monoxide in toluene

 

C  0.000000 0   0.000000 0   0.000000 0

O  1.173000 0   0.000000 0   0.000000 0

 

In ethanol, the first excited-state energy is 6.739 eV:

 

  CI trans.  energy frequency wavelength oscillator ---------polarization---------    dipole   ------components-----

 st.  symm.    eV      cm-1       nm      strength       x          y         z       moment     x       y       z

 

   2      6.7386887    54351.    183.99  0.000000                                   2.375820   2.376  -0.000  -0.000

 

In toluene, the first excited-state energy is 6.652 eV:

 

  CI trans.  energy frequency wavelength oscillator ---------polarization---------    dipole   ------components-----

 st.  symm.    eV      cm-1       nm      strength       x          y         z       moment     x       y       z

 

   2      6.6519378    53652.    186.39  0.000000                                   2.432843   2.433  -0.000  -0.000

 

The computed solvatochromic shift is (6.739 eV - 6.652 eV) = 0.087 eV.

(3) Detailed analysis of INDO/CI Excited States

 

The INDO/S Hamiltonian (or equivalently, ZINDO/S), has parameters designed specifically to reproduce vertical excited-state energies using a configuration interaction (CI) approach with single excitations (CIS), and has also shown success in understanding excited states with large double-excitation character using either single and double excitations (CISD) or a multi-reference approach (MRCI).

 

This job illustrates how to compute the excited states of all-trans-octatetraene using INDO/S with an MRCI approach. The reference determinants are generated using a complete active space approach within the C.A.S.=(2,1) active space, which generates three reference determinants: (1) the SCF ground state, (2) the single HOMO → LUMO excitation, and (3) the double HOMO, HOMO → LUMO, LUMO excitation. MRCI then uses these three references determinants as starting points to generate single excitations within the C.I.=(24,12) active space. In this example, we limit the CI active space to the 500 lowest-energy electron configurations within this active space (MAXCI=500), print the lowest 50 excited states (WRTCI=50), and print only configurations that contribute to each excited state with coefficients larger than 0.1 (WRTCONF=0.1).

 

The input file is:

 

INDO MRCI C.I.=(24,12) C.A.S.=(2,1) MAXCI=500 WRTCI=50 WRTCONF=0.1

all-trans-octatetraene

 

  C       0.136428 0       4.312612 0       0.000000 0

  C       0.650845 0       3.070957 0       0.000000 0

  C      -0.136428 0       1.855969 0       0.000000 0

  C       0.389263 0       0.606310 0       0.000000 0

  C      -0.389263 0      -0.606310 0       0.000000 0

  C       0.136428 0      -1.855969 0       0.000000 0

  C      -0.650845 0      -3.070957 0       0.000000 0

  C      -0.136428 0      -4.312612 0       0.000000 0

  H       0.770425 0       5.193819 0       0.000000 0

  H      -0.937761 0       4.483413 0       0.000000 0

  H       1.734237 0       2.945379 0       0.000000 0

  H      -1.221062 0       1.972790 0       0.000000 0

  H       1.474577 0       0.494288 0       0.000000 0

  H      -1.474577 0      -0.494288 0       0.000000 0

  H       1.221062 0      -1.972790 0       0.000000 0

  H      -1.734237 0      -2.945379 0       0.000000 0

  H       0.937761 0      -4.483413 0       0.000000 0

  H      -0.770425 0      -5.193819 0       0.000000 0

 

The output file contains a statement of which electron configurations are used in the CI calculation:

 

 Reference determinate nber   1 is ALPHA 1 -  21

 CAS excitations amongst orbitals:  21  22

 SINGLE excitations FROM orbs  10 to  21 INTO orbs  22 to  33

 

Even though there are 531 possible electron configurations with singlet spin within this active space, the MAXCI=500 keyword limits the CI matrix to the 500 lowest-energy electron configurations within this active space. The output file contains the list of all 500 configurations within the CI matrix and their energies, which include the SCF ground state, single excitations, double excitations (generated as single excitations relative to the HOMO → LUMO reference determinant), and triple excitations (generated as single excitations relative to the HOMO, HOMO → LUMO, LUMO reference determinant). For example, the pure HOMO → LUMO excitation (21→22) has an energy of 4.256 eV relative to the SCF ground state:

 

CI excitations=  500:        =500

 The lowest  500 spin-adapted configurations of multiplicity=  1

     sym   eV   cm**-1 -dets- dipole oscilator X FRAG ....Excitations named from first reference determinate

                       tot  #  Debye  strength

 

   1      0.000      0.  1  1  0.0000 0.000000 0  0 (            )->(               )

   2      4.256  34328.  2  1  0.0000 1.853465 1  1 (  21        )->(  22           )

   3      5.592  45100.  2  1  0.0000 0.000000 1  1 (  21        )->(  23           )

   4      6.128  49424.  2  1  0.0000 0.000000 1  1 (  20        )->(  22           )

   5      6.195  49965.  2  1  0.0000 0.000013 1  1 (  21        )->(  26           )

   6      6.267  50544.  1  1  0.0000 0.000000 2  0 (  21  21    )->(  22  22       )

   7      6.496  52392.  2  1  0.0000 0.010230 1  1 (  21        )->(  24           )

...

  95     12.917 104186.  2  1  0.0000 0.000000 3  1 (  20  21  21)->(  22  22  23   )

 

The output file then contains the states obtained by diagonalizing the CI matrix. In this case, the ground state is stabilized by 8657 cm^-1 relative to the SCF ground state:

 

Depression of ground-state after CI=      -8657. cm**-1 Energy= -1399.9520161 eV

 

The first excited state (state 2) is 4.668 eV higher in energy than the ground state, and the second excited state (state 3) is 4.890 eV higher in energy than the ground state. The first excited state is a dark state (oscillator strength = 0) and the second excited state is bright (oscillator strength = 1.130) with its transition dipole moment aligned almost perfectly along the y axis. This is in good agreement with the experimental properties of this molecule, which has a dark first excited state slightly lower in energy than the first bright state.

 

  CI trans.  energy frequency wavelength oscillator ---------polarization---------    dipole   ------components-----

 st.  symm.    eV      cm-1       nm      strength       x          y         z       moment     x       y       z

 

   2      4.6680976    37651.    265.60  0.000000                                   0.000000   0.000   0.000  -0.000

   3      4.8897203    39438.    253.56  1.130960  -0.039398   0.999224  -0.000000  0.000000  -0.000  -0.000  -0.000

 

After the summary list of excited states, the output file contains the list of electron configurations that contribute to each excited state. The ground state (State 1) contains 85% electron configuration 1, which corresponds to the SCF ground state (numbering corresponds to the numbering in the list of all 500 electron configurations printed earlier). The ground state also contains 7% electron configuration 6 (21, 21 → 22, 22) and smaller contributions from other electron configurations.

 

State    1  0.0000     CI coeff  CI percent

    Config       1   0.92336963  0.85261147

    Config       6  -0.27185071  0.07390281

    Config      44  -0.11288867  0.01274385

    Config      85   0.13615963  0.01853945

    Config     116  -0.10641287  0.01132370

 Total coeff printed             0.99485866

 

The first excited state has three major contributions: configurations 3 (21 → 23), 4 (20 → 22), and 6 (21, 21 → 22, 22). This state is the first excited-state only because of the mixing of a doubly excited configuration with single excitations; in a CIS calculation, this excited state will be higher in energy than the first bright state.

 

State    2  4.6681     CI coeff  CI percent

    Config       1  -0.24379600  0.05943649

    Config       3   0.54559432  0.29767316

    Config       4  -0.43522910  0.18942437

    Config       6  -0.51352715  0.26371013

    Config      18   0.11118656  0.01236245

    Config      33   0.16400700  0.02689830

    Config      44  -0.26248831  0.06890011

    Config      56   0.12774098  0.01631776

    Config     116  -0.11973570  0.01433664

 Total coeff printed             0.94905942

 

The second excited state has only one major contribution: configuration 2 (21 → 22).

 

State    3  4.8897     CI coeff  CI percent

    Config       2   0.97454714  0.94974213

 Total coeff printed             0.94974213