Tutorial on Energetic Analysis with NBO Deletions ($DEL Keylist)

Introduction to the $DEL Keylist and NBO Energetic Analysis

In G9X and GAMESS implementations, the NBO program makes it possible to determine the energetic effect of deleting certain NBOs, groups of NBOs, or specific NBO donor-acceptor interactions. This is done by including (following the main $NBO keylist) a special "deletions" $DEL keylist of the form

$DEL keyword(s) $END

(For a G9X host program, the POP=NBODEL keyword must also be included in the route card.)

As described in the NBO 5.0 Manual, a total of nine deletion types (each with specific keyword syntax) can be used to select specific classes of intra- or intermolecular Fock matrix elements for deletion. When combined with geometry optimization, the $DEL keylist also makes it possible to determine the structural consequences of specific NBO donor-acceptor interactions as the difference between the original fully-optimized geometry and the $DEL-optimized geometry.

In this Tutorial we employ $DEL keyword options to illustrate how simple chemical questions may be addressed in the case of a configurational energy difference.

Cis vs. Trans Configuration of Difluoroethene

1,2-difluoroethene exists in cis and trans isomers



On the basis of steric and electrostatic factors the trans isomer is expected to be significantly more stable, because the two bulky F atoms and the repelling negative charges of the two CF bond dipoles are maximally separated in this configuration. However, the cis isomer is actually found to be very similar in energy. For example, in a simple RHF/6-31G* model with idealized trigonal geometry (RCF = 1.33 A, RCC = 1.34 A, RCH = 1.08 A) the calculated energies are

Ecis = -275.71726 a.u.,
Etrans = -275.71761 a.u.

corresponding to only a slight energy difference favoring the trans isomer.

Ecis - Etrans = 0.22 kcal/mol

The model RHF calculation exhibits the near-equivalence of cis and trans configurational energies but does not help us to understand why the cis configuration is so competitive. In this Tutorial we show how to address such questions by using $DEL keylists to investigate the effect of specific NBO donor-acceptor interactions on the isomeric energy difference.

Can the Surprising Stability of the Cis Isomer be Attributed to Electronic Delocalization?

The role of electronic delocalization can be quantitatively assessed by deleting all non-Lewis (starred) NBOs from the basis set with the NOSTAR keyword

$DEL NOSTAR $END

The resulting "natural Lewis structure" wavefunction is perfectly localized, with all Lewis-type NBOs doubly occupied. By the variational principle, the Lewis-type wavefunction has an energy E(L) that is higher than the original energy E(full). The net energy difference E(NL)

E(NL) = E(full) - E(L)

gives the stabilizing effect of the delocalizing (non-Lewis) contributions.

For the cis isomer, the sample G9X input file

 %mem=2000000                                         
 #N HF/6-31G*  NOSYMM POP=NBODel

 CHF::CHF (cis)

   0  1
  C
  C   1   1.34
  F   1   1.33   2  120.0
  F   2   1.33   1  120.0  3    0.0
  H   1   1.08   2  120.0  3  180.0
  H   2   1.08   1  120.0  4  180.0

 $NBO file=dfe_c $END
 $DEL
    nostar
 $END

leads to the $DEL output shown below. The first output segment identifies the type of deletion ("NOSTAR: Delete all Rydberg/antibond NBOs") and lists the 48 Rydberg and antibond NBOs (17-64) that were deleted:

 NOSTAR: Delete all Rydberg/antibond NBOs
 Deletion of the following orbitals from the NBO Fock matrix:
   17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36

   37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56

   57  58  59  60  61  62  63  64

The next segment lists the NBOs with their original occupancies ("No deletions") and current occupancies ("This deletion"), together with the net change in each:

 Orbital occupancies:

       Orbital                   No deletions   This deletion   Change
 ------------------------------------------------------------------------------
   1. BD ( 1) C 1- C 2               1.99763        2.00000    0.00237
   2. BD ( 2) C 1- C 2               1.99676        2.00000    0.00324
   3. BD ( 1) C 1- F 3               1.99702        2.00000    0.00298
   4. BD ( 1) C 1- H 5               1.98669        2.00000    0.01331
   5. BD ( 1) C 2- F 4               1.99702        2.00000    0.00298
   6. BD ( 1) C 2- H 6               1.98669        2.00000    0.01331
   7. CR ( 1) C 1                    1.99873        2.00000    0.00127
   8. CR ( 1) C 2                    1.99873        2.00000    0.00127
   9. CR ( 1) F 3                    1.99994        2.00000    0.00006
  10. CR ( 1) F 4                    1.99994        2.00000    0.00006
  11. LP ( 1) F 3                    1.99224        2.00000    0.00776
  12. LP ( 2) F 3                    1.97331        2.00000    0.02669
  13. LP ( 3) F 3                    1.94914        2.00000    0.05086
  14. LP ( 1) F 4                    1.99224        2.00000    0.00776
  15. LP ( 2) F 4                    1.97331        2.00000    0.02669
  16. LP ( 3) F 4                    1.94914        2.00000    0.05086
  17. RY*( 1) C 1                    0.00613        0.00000   -0.00613
  18. RY*( 2) C 1                    0.00453        0.00000   -0.00453
  19. RY*( 3) C 1                    0.00197        0.00000   -0.00197
  20. RY*( 4) C 1                    0.00131        0.00000   -0.00131
  21. RY*( 5) C 1                    0.00026        0.00000   -0.00026
  22. RY*( 6) C 1                    0.00017        0.00000   -0.00017
  23. RY*( 7) C 1                    0.00008        0.00000   -0.00008
  24. RY*( 8) C 1                    0.00000        0.00000    0.00000
  25. RY*( 9) C 1                    0.00001        0.00000   -0.00001
  26. RY*(10) C 1                    0.00001        0.00000   -0.00001
  27. RY*( 1) C 2                    0.00613        0.00000   -0.00613
  28. RY*( 2) C 2                    0.00453        0.00000   -0.00453
  29. RY*( 3) C 2                    0.00197        0.00000   -0.00197
  30. RY*( 4) C 2                    0.00131        0.00000   -0.00131
  31. RY*( 5) C 2                    0.00026        0.00000   -0.00026
  32. RY*( 6) C 2                    0.00017        0.00000   -0.00017
  33. RY*( 7) C 2                    0.00008        0.00000   -0.00008
  34. RY*( 8) C 2                    0.00000        0.00000    0.00000
  35. RY*( 9) C 2                    0.00001        0.00000   -0.00001
  36. RY*(10) C 2                    0.00001        0.00000   -0.00001
  37. RY*( 1) F 3                    0.00171        0.00000   -0.00171
  38. RY*( 2) F 3                    0.00159        0.00000   -0.00159
  39. RY*( 3) F 3                    0.00007        0.00000   -0.00007
  40. RY*( 4) F 3                    0.00004        0.00000   -0.00004
  41. RY*( 5) F 3                    0.00001        0.00000   -0.00001
  42. RY*( 6) F 3                    0.00001        0.00000   -0.00001
  43. RY*( 7) F 3                    0.00000        0.00000    0.00000
  44. RY*( 8) F 3                    0.00000        0.00000    0.00000
  45. RY*( 9) F 3                    0.00000        0.00000    0.00000
  46. RY*(10) F 3                    0.00001        0.00000   -0.00001
  47. RY*( 1) F 4                    0.00171        0.00000   -0.00171
  48. RY*( 2) F 4                    0.00159        0.00000   -0.00159
  49. RY*( 3) F 4                    0.00007        0.00000   -0.00007
  50. RY*( 4) F 4                    0.00004        0.00000   -0.00004
  51. RY*( 5) F 4                    0.00001        0.00000   -0.00001
  52. RY*( 6) F 4                    0.00001        0.00000   -0.00001
  53. RY*( 7) F 4                    0.00000        0.00000    0.00000
  54. RY*( 8) F 4                    0.00000        0.00000    0.00000
  55. RY*( 9) F 4                    0.00000        0.00000    0.00000
  56. RY*(10) F 4                    0.00001        0.00000   -0.00001
  57. RY*( 1) H 5                    0.00061        0.00000   -0.00061
  58. RY*( 1) H 6                    0.00061        0.00000   -0.00061
  59. BD*( 1) C 1- C 2               0.02181        0.00000   -0.02181
  60. BD*( 2) C 1- C 2               0.09725        0.00000   -0.09725
  61. BD*( 1) C 1- F 3               0.01035        0.00000   -0.01035
  62. BD*( 1) C 1- H 5               0.01735        0.00000   -0.01735
  63. BD*( 1) C 2- F 4               0.01035        0.00000   -0.01035
  64. BD*( 1) C 2- H 6               0.01735        0.00000   -0.01735

Note that this deletion has resulted in all Lewis-type NBOs 1-16 becoming doubly occupied and all remaining non-Lewis-type NBOs becoming empty, the idealized Lewis structure limit.

The next output segment displays the calculated energy of the deletion and the energy change (in atomic units and kcal/mol):

 NEXT STEP:  Evaluate the energy of the new density matrix
             that has been constructed from the deleted NBO
             Fock matrix by doing one SCF cycle.

 ------------------------------------------------------------------------------
 Warning!  Cutoffs for single-point calculations used.
 Requested convergence on RMS density matrix=1.00D-04 within   1 cycles.
 Requested convergence on MAX density matrix=1.00D-02.
 Requested convergence on             energy=5.00D-05.
 >>>>>>>>>> Convergence criterion not met.
 SCF Done:  E(RHF) =  -275.459598747     A.U. after    2 cycles
             Convg  =    0.9196D-02             -V/T =  2.0040
             S**2   =   0.0000

 ------------------------------------------------------------------------------
   Energy of deletion :       -275.459598747
     Total SCF energy :       -275.717261115
                          -------------------
        Energy change :          0.257662 a.u.,         161.690 kcal/mol
 ------------------------------------------------------------------------------

[Note that the G9X program always prints a "Convergence criterion not met" warning message, because the deleted density does not exactly match the starting density in the one-pass SCF evaluator method used by $DEL. This message can be safely ignored. For the NOSTAR case, the energy of deletion is a strict variational expectation value of the natural Lewis structure wavefunction, but for other deletions a slight variational inconsistency is incurred due to the slight difference between the deleted density distribution and the final density from which the converged Fock operator was constructed.]

From the above output, we can recognize that

Ecis(full) = -275.71726 a.u.
Ecis(L) = -275.45960 a.u.
Ecis(NL) = -0.25762 a.u.

From the $DEL output for the trans isomer we obtain similarly

Etrans(full) = -275.71761 a.u.
Etrans(L) = -275.46494 a.u.
Etrans(NL) = -0.25268 a.u.

From these values we find

Ecis(L) - Etrans(L) = +0.00534 a.u. = +3.34 kcal/mol
Ecis(NL) - Etrans(NL) = -0.00494 a.u. = -3.11 kcal/mol

Thus, the localized E(L) contribution favors the trans configuration by >3 kcal/mol, in accordance with the expected steric and electrostatic differences. However, the small delocalization contribution is seen to strongly favor the cis isomer by a similar amount, leading to nearly equal total energies for the two isomers. The surprising stability of the cis isomer can therefore be attributed to the electronic delocalization energy E(NL).

What Specific NBO Donor-Acceptor Interactions are Responsible for this Preference?

To further dissect E(NL) into specific donor-acceptor interactions, we can consider more selective $DEL keylists of the form

$DEL
delete n elements
  d1 a1
  d2 a2
   . . .
  dn an
$END

where each di is the number of a Lewis-type donor NBO and ai is the number of a non-Lewis-type acceptor NBO. For example, the possible vicinal hyperconjugative delocalizations between the CF bond (NBO 3) and CH bond (NBO 4) on C(1) with the CF antibond (NBO 63) and CH antibond (NBO 64) on C(2) could be specified with

$DEL
delete 4 elements
  3  63
  3  64
  4  63
  4  64
$END

By such exploratory deletions, one can verify that donor-acceptor interactions involving the Rydberg-type (RY*) NBOs 17-58 have negligible effect on the cis/trans energy difference. Similarly, interactions involving the acceptor antibond (BD*) NBOs 59-64 are only significant when paired with vicinal donor bond (BD) NBOs on the opposite carbon atom.

The most important contributions to the cis-trans energy difference are found to be the two equivalent vicinal interactions involving CH bonds as donors (NBOs 4,6) and CF* antibonds as acceptors (NBOs 61,63). The 4-63 interaction, e.g., can be deleted with the $DEL keylist

$DEL
delete 1 element
  4  63
$END

The deletion energy for each such CH-CF* interaction is found to be 6.02 kcal/mol in the cis isomer but only 1.22 kcal/mol in the trans isomer, thus contributing strongly to stabilizing the cis isomer. (The calculated deletion energies agree sensibly with the corresponding 2nd-order perturbation estimates of 6.92 and 1.44 kcal/mol.)

Why do vicinal CH-CF* interactions favor the cis isomer? Each such CH-CF* interaction is antiperiplanar in the cis isomer, but synperiplanar in the trans isomer. As simple NBO overlap diagrams make clear, this delocalizing interaction is considerably stronger in the anti arrangement than in the syn arrangement.



Thus, the cis isomer benefits from two strong antiperiplanar CH-CF* delocalizations, whereas the trans isomer has none. [There is evidently a similar syn/anti difference between CH-CH* interactions that favors the trans isomer, but the CH-CF* interactions are the more important due to the greater acceptor strength of the CF* antibond.] We can therefore conclude that vicinal CH-CF* donor-acceptor interactions are most responsible for stabilizing the cis isomer.

What Influences Do Hyperconjugative Delocalizations Exert on Other Geometrical Variables?

From the NBO overlap diagrams shown above, we can visualize how slight distortions of geometry might further strengthen the donor-acceptor interactions. In the cis isomer, for example, opening up the C-C-F angle or closing down the C-C-H angle will both increase the favorable overlap between the backside lobe of the CF* antibond and the shoulder of the CH bond to allow stronger delocalization and stabilization, as will shortening the central C-C bond. Such distortions would naturally be opposed by the skeletal bonding framework, as the bonding hybrids strive to maintain optimal orientations and separations for strong covalent bonding. Given the weakness of hyperconjugative donor-acceptor delocalizations compared to covalent bonding interactions, we expect that the hyperconjugative effects on molecular geometry are small but detectable (perhaps a few degrees or a few hundredths of an Angstrom). How can we quantitate these hyperconjugative distortions?

A $DEL keylist combined with geometry optimization allows one to determine what the optimal geometry would be in the absence of a particular hyperconjugative interaction. As a simple example, let us consider how the geometry of difluoroethene would be altered if the two CF* antibonds were not present, so that no hyperconjugative delocalizations involving these orbitals were possible. We can address this question by re-optimizing the geometry with the two CF* antibonds (NBOs 63, 64) deleted. The input file to perform such a $DEL-optimization for the cis isomer is shown below.

 %mem=2000000                                      
 #N HF/6-31G*  NOSYMM POP=NBODel OPT

 CHF::CHF (cis)

   0  1
  C
  C   1   1.34
  F   1   1.33   2  120.0
  F   2   1.33   1  120.0  3    0.0
  H   1   1.08   2  120.0  3  180.0
  H   2   1.08   1  120.0  4  180.0

 $NBO file=dfe_c $END
 $DEL
 delete 2 orbitals 
   63 64 
 $END

In this job, the G9X geometry optimizer searches for an equilibrium structure on the E($DEL) surface, which differs from the E(full) surface by deletion of contributions from NBOs 63, 64. Removing the "POP=NBODel" keyword and repeating the optimization yields the usual full-basis optimized structure for comparison. Some bond angles and distances for the E(full) and E($DEL) optimized geometries are shown in the table below:

Optimized Geometry of Cis Isomer

Geometry Parameter Full $DEL diff
bond length (Å) C-C 1.307 1.333 -0.026
C-F 1.324 1.326 -0.002
C-H 1.070 1.068 +0.002
bond angle (°) C-C-F 122.6 120.6 +2.0
C-C-H 123.0 127.1 -4.1

From these comparisons one can see that inclusion of CF* delocalizations has three primary effects on the molecular geometry: (i) significant shortening (by 0.026 A) of the central C-C bond; (ii) significant opening (by 2.0 degrees) of the C-C-F bond angle; (iii) significant closing (by 4.1 degrees) of the C-C-H bond angle. As noted above, all three of these changes are expected from the general form of the CH-CF* orbital overlap diagram.


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