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
(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.
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
corresponding to only a slight energy difference favoring the trans isomer.
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.
The role of electronic delocalization can be quantitatively assessed by deleting all non-Lewis (starred) NBOs from the basis set with the NOSTAR keyword
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)
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
From the $DEL output for the trans isomer we obtain similarly
From these values we find
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).
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.
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:
| 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.