When interfaced to Gaussian or GAMESS, 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 Gaussian, the POP=NBO6DEL keyword (for Rev D) or POP=NBODEL keyword (for pre-Rev D) must also be included on the route card.]

The NBO Manual (Section B.5) describes nine deletion types (each with specific keyword syntax) that 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:

Of the two, the *trans* isomer is expected to be significantly
more stable, on the basis of steric and electrostatic factors, 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, for a simple RHF/6-31G* model with idealized
trigonal geometry (*R*_{CF} = 1.33°,
*R*_{CC} = 1.34°,
*R*_{CH} = 1.08°) 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 NBOs from the basis set with
the LEWIS 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 Gaussian input file

# rhf/6-31g* pop=nbo6del nosymm cis-1,2-difluoroethene...rhf/6-31g*//idealized geometry 0 1 C C 1 1.34 F 1 1.33 2 120. F 2 1.33 1 120. 3 0. 0 H 1 1.08 2 120. 3 180. 0 H 2 1.08 1 120. 4 180. 0 $nbo $end $del lewis $end

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

LEWIS: Delete all non-Lewis 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. CR ( 1) C 1 1.99872 2.00000 0.00128 2. CR ( 1) C 2 1.99872 2.00000 0.00128 3. CR ( 1) F 3 1.99994 2.00000 0.00006 4. CR ( 1) F 4 1.99994 2.00000 0.00006 5. LP ( 1) F 3 1.99225 2.00000 0.00775 6. LP ( 2) F 3 1.97329 2.00000 0.02671 7. LP ( 3) F 3 1.94905 2.00000 0.05095 8. LP ( 1) F 4 1.99225 2.00000 0.00775 9. LP ( 2) F 4 1.97329 2.00000 0.02671 10. LP ( 3) F 4 1.94905 2.00000 0.05095 11. BD ( 1) C 1- C 2 1.99758 2.00000 0.00242 12. BD ( 2) C 1- C 2 1.99676 2.00000 0.00324 13. BD ( 1) C 1- F 3 1.99680 2.00000 0.00320 14. BD ( 1) C 1- H 5 1.98730 2.00000 0.01270 15. BD ( 1) C 2- F 4 1.99680 2.00000 0.00320 16. BD ( 1) C 2- H 6 1.98730 2.00000 0.01270 17. BD*( 1) C 1- C 2 0.02163 0.00000 -0.02163 18. BD*( 2) C 1- C 2 0.09741 0.00000 -0.09741 19. BD*( 1) C 1- F 3 0.00987 0.00000 -0.00987 20. BD*( 1) C 1- H 5 0.01756 0.00000 -0.01756 21. BD*( 1) C 2- F 4 0.00987 0.00000 -0.00987 22. BD*( 1) C 2- H 6 0.01756 0.00000 -0.01756 23. RY ( 1) C 1 0.00612 0.00000 -0.00612 24. RY ( 2) C 1 0.00453 0.00000 -0.00453 25. RY ( 3) C 1 0.00198 0.00000 -0.00198 26. RY ( 4) C 1 0.00131 0.00000 -0.00131 27. RY ( 5) C 1 0.00026 0.00000 -0.00026 28. RY ( 6) C 1 0.00017 0.00000 -0.00017 29. RY ( 7) C 1 0.00008 0.00000 -0.00008 30. RY ( 8) C 1 0.00001 0.00000 -0.00001 31. RY ( 9) C 1 0.00000 0.00000 0.00000 32. RY (10) C 1 0.00000 0.00000 0.00000 33. RY ( 1) C 2 0.00612 0.00000 -0.00612 34. RY ( 2) C 2 0.00453 0.00000 -0.00453 35. RY ( 3) C 2 0.00198 0.00000 -0.00198 36. RY ( 4) C 2 0.00131 0.00000 -0.00131 37. RY ( 5) C 2 0.00026 0.00000 -0.00026 38. RY ( 6) C 2 0.00017 0.00000 -0.00017 39. RY ( 7) C 2 0.00008 0.00000 -0.00008 40. RY ( 8) C 2 0.00001 0.00000 -0.00001 41. RY ( 9) C 2 0.00000 0.00000 0.00000 42. RY (10) C 2 0.00000 0.00000 0.00000 43. RY ( 1) F 3 0.00171 0.00000 -0.00171 44. RY ( 2) F 3 0.00159 0.00000 -0.00159 45. RY ( 3) F 3 0.00007 0.00000 -0.00007 46. RY ( 4) F 3 0.00004 0.00000 -0.00004 47. RY ( 5) F 3 0.00002 0.00000 -0.00002 48. RY ( 6) F 3 0.00001 0.00000 -0.00001 49. RY ( 7) F 3 0.00000 0.00000 0.00000 50. RY ( 8) F 3 0.00000 0.00000 0.00000 51. RY ( 9) F 3 0.00000 0.00000 0.00000 52. RY (10) F 3 0.00000 0.00000 0.00000 53. RY ( 1) F 4 0.00171 0.00000 -0.00171 54. RY ( 2) F 4 0.00159 0.00000 -0.00159 55. RY ( 3) F 4 0.00007 0.00000 -0.00007 56. RY ( 4) F 4 0.00004 0.00000 -0.00004 57. RY ( 5) F 4 0.00002 0.00000 -0.00002 58. RY ( 6) F 4 0.00001 0.00000 -0.00001 59. RY ( 7) F 4 0.00000 0.00000 0.00000 60. RY ( 8) F 4 0.00000 0.00000 0.00000 61. RY ( 9) F 4 0.00000 0.00000 0.00000 62. RY (10) F 4 0.00000 0.00000 0.00000 63. RY ( 1) H 5 0.00061 0.00000 -0.00061 64. RY ( 1) H 6 0.00061 0.00000 -0.00061

Note that this deletion has resulted in all Lewis-type NBOs (1-16) becoming doubly occupied and all remaining non-Lewis-type NBOs (17-64) 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: Perform one SCF cycle to evaluate the energy of the new density matrix constructed from the deleted NBO Fock matrix. ------------------------------------------------------------------------------ . . . Requested convergence on RMS density matrix=1.00D-08 within 1 cycles. Requested convergence on MAX density matrix=1.00D-06. Requested convergence on energy=1.00D-06. No special actions if energy rises. Fock symm off for IB=2 I1= 1 I= 25 J= 14 Cut=1.00D-07 Err=3.63D-02 Fock matrix is not symmetric: symmetry in diagonalization turned off. >>>>>>>>>> Convergence criterion not met. SCF Done: E(RHF) = -275.461162154 A.U. after 2 cycles NFock= 1 Conv=0.92D-02 -V/T= 2.0040 . . . ------------------------------------------------------------------------------ Energy of deletion : -275.461162154 Total SCF energy : -275.717263936 ------------------- Energy change : 0.256102 a.u., 160.706 kcal/mol ------------------------------------------------------------------------------

[Note that the Gaussian 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 LEWIS 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.]

[Also note that the "nosymm" keyword is added to the route card of the Gaussian input file for deletions calculation to override the use of symmetry in the SCF energy evaluation. This keyword can be safely removed if the deletions to be performed ensure that the deleted density retains the symmetry of the fully delocalized SCF density. In fact, the Lewis deletions described here do preserve symmetry, so "nosymm" could be discarded. However, several of the deletions described below do not preserve symmetry. If uncertain, include "nosymm" on the Gaussian route card. Additional comments on symmetry can be found in Section B.5 of the NBO Manual.]

From the above output, we recognize that

Similarly, from the $DEL output for the *trans* isomer, we
obtain

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

$DEL deletenelementsd1 a1d2 a2. . .dn an$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 13) and CH bond (NBO 14) on C(1) with the CF antibond
(NBO 21) and CH antibond (NBO 22) on C(2) could be specified with

$DEL delete 4 elements 13 21 13 22 14 21 14 22 $END

By such exploratory deletions, one can verify that donor-acceptor
interactions involving the Rydberg-type (RY) NBOs 23-64 have
negligible effect on the *cis/trans* energy difference.
Similarly, interactions involving the acceptor antibond (BD*) NBOs
17-22 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 14,16) and CF* antibonds as acceptors
(NBOs 19,21). The 14-21 interaction, e.g., can be deleted with the
$DEL keylist

$DEL delete 1 element 14 21 $END

The deletion energy for each such CH-CF* interaction is found to be
5.97 kcal/mol in the *cis* isomer but only 1.19 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 perturbative estimates of
6.86 and 1.40 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*.

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 19,21) deleted. The G09 input file to perform such a
$DEL-optimization for the *cis* isomer is shown below. (The
corresponding input file for GAMESS is given at the end of this
tutorial.)

# rhf/6-31g* pop=nbo6del nosymm opt Geometry optimization of cis-1,2-difluoroethene (no CF*s) 0 1 C C 1 CC F 1 CF 2 CCF F 2 CF 1 CCF 3 0. 0 H 1 CH 2 CCH 3 180. 0 H 2 CH 1 CCH 4 180. 0 CC 1.34 CF 1.33 CH 1.08 CCF 120. CCH 120. $nbo $end $choose lone 3 3 4 3 end bond d 1 2 s 1 3 s 1 5 s 2 4 s 2 6 end $end $del delete 2 orbitals 19 21 $end

In this job, the Gaussian geometry optimizer searches for an
equilibrium structure on the *E*($DEL) surface, which differs
from the *E*(full) surface by deletion of contributions from NBOs
19,21. Removing the POP=NBO6DEL keyword and repeating the
optimization yields the usual full-basis optimized structure for
comparison. Some bond angles and distances for the *E*(full)
and *E*($DEL) optimized geometries are shown in the table
below:

Geometry Parameter | full | $DEL | diff | |
---|---|---|---|---|

bond length (Å) | C-C | 1.307 | 1.342 | -0.035 |

C-F | 1.324 | 1.308 | +0.016 | |

C-H | 1.070 | 1.070 | 0.000 | |

bond angle (°) | C-C-F | 122.6 | 117.9 | +4.7 |

C-C-H | 123.0 | 127.8 | -4.8 |

From these comparisons one can see that delocalizations into the CF* antibonds have three primary effects on molecular geometry: (i) significant shortening (by 0.035 A) of the central C-C bond length; (ii) significant opening (by 4.7 degrees) of the C-C-F bond angle; (iii) significant closing (by 4.8 degrees) of the C-C-H bond angle. As noted above, all three of these changes are expected to enhance CH-CF* delocalization by strengthening donor-acceptor overlap.

Similar optimizations of the *trans* isomer reveal that CF*
delocalizations shorten the C-C bond length but have comparatively
limited influence on the C-C-F and C-C-H bond angles:

Geometry Parameter | full | $DEL | diff | |
---|---|---|---|---|

bond length (Å) | C-C | 1.307 | 1.340 | -0.033 |

C-F | 1.329 | 1.308 | +0.021 | |

C-H | 1.070 | 1.072 | -0.002 | |

bond angle (°) | C-C-F | 120.2 | 121.0 | -0.8 |

C-C-H | 125.3 | 124.5 | +0.8 |

[Note that we recommend using the directed NBO search ($CHOOSE
input) when performing geometry optimizations with deletions. These
optimizations are based on numerical gradients, and thereby sample a
large number of geometries as *E*($DEL) is minimized. $CHOOSE
input ensures that NBO uses the same Lewis structure at each geometry.
Because the orbital numbers used to specify the deletion are tied
directly to a particular Lewis structure, changes in the Lewis structure
from geometry to geometry are bound to confuse the optimizer. $CHOOSE
input is described in Section B.4 of the NBO Manual.]

The following is the corresponding input file for GAMESS for 1,2-cis-difluoroethene.

$contrl scftyp=rhf runtyp=trudge coord=hint $end $basis gbasis=n31 ngauss=6 ndfunc=1 $end $trudge optmiz=geometry npar=5 iex(1)=11,21,22,31,32 $end $trurst tolf=0.0001 tolr=0.001 fnoise=0.00001 $end $nbo $end $choose lone 3 3 4 3 end bond d 1 2 s 1 3 s 1 5 s 2 4 s 2 6 end $end $del delete 2 orbitals 19 21 $end $data Geometry optimization of cis-1,2-difluoroethene (no CF*s) Cnv 2 C 6. LC 0.67 0.00 0.00 + O I F 9. TCT 1.33 120.00 0.00 + 1 O K H 1. TCT 1.08 120.00 180.00 + 1 O K $end

Note that the calculation uses GAMESS' TRUDGE numerical optimizer and that the geometry must be specified in Hildebrandt (coord=hint) internal coordinates. See the GAMESS manual for information on runtyp=trudge, coord=hint, and the $trudge and $trurst groups. GAMESS optimizes the cis isomer to the same $DEL geometry described above.