10.12.3 Linear-Scaling NMR Chemical Shift Calculations‣ 10.12 NMR and Other Magnetic Properties ‣ Chapter 10 Molecular Properties and Analysis ‣ Q-Chem 6.0 User’s Manual

In conventional implementations, the cost for computation of NMR chemical shifts within even the simplest quantum chemical methods such as Hartree-Fock of DFT increases cubically with molecular size $M$ , ${\cal{O}}({M^{3}})$ . As such, NMR chemical shift calculations have largely been limited to molecular systems on the order of 100 atoms, assuming no symmetry. For larger systems it is crucial to reduce the increase of the computational effort to linear, which is possible for systems with a nonzero HOMO/LUMO gaps and was reported for the first time by Kussmann and Ochsenfeld. ⁸⁷⁹ Ochsenfeld C., Kussmann J., Koziol F.
Angew. Chem.
(2004), 116, pp. 4585. Link ^, ⁶⁵⁰ Kussmann J., Ochsenfeld C.
J. Chem. Phys.
(2007), 127, pp. 054103. Link This approach incurs no loss of accuracy with respect to traditional cubic-scaling implementations, and makes feasible NMR chemical shift calculations using Hartree-Fock or DFT approaches in molecular systems with 1,000+ atoms. For many molecular systems the Hartree-Fock (GIAO-HF) approach provides typically an accuracy of 0.2–0.4 ppm for the computation of ${}^{1}$ H NMR chemical shifts, for example. ⁸⁸² Ochsenfeld C.
Phys. Chem. Chem. Phys.
(2000), 2, pp. 2153. Link ^, ⁸⁷⁶ Ochsenfeld C. et al.
J. Am. Chem. Soc.
(2001), 123, pp. 2597. Link ^, ¹⁴² Brown S. P. et al.
Angew. Chem. Int. Ed. Engl.
(2001), 40, pp. 717. Link ^, ⁸⁷⁸ Ochsenfeld C. et al.
Solid State Nucl. Mag.
(2002), 22, pp. 128. Link ^, ⁸⁷⁹ Ochsenfeld C., Kussmann J., Koziol F.
Angew. Chem.
(2004), 116, pp. 4585. Link GIAO-HF/6-31G* calculations with 1,003 atoms and 8,593 basis functions, without symmetry, have been reported. ⁸⁷⁹ Ochsenfeld C., Kussmann J., Koziol F.
Angew. Chem.
(2004), 116, pp. 4585. Link GIAO-DFT calculations are even simpler and faster for density functionals that do not contain Hartree-Fock exchange.

The present implementation of NMR shieldings employs the LinK (linear exchange, “K”) method ⁸⁸⁰ Ochsenfeld C., White C. A., Head-Gordon M.
J. Chem. Phys.
(1998), 109, pp. 1663. Link ^, ⁸⁸¹ Ochsenfeld C.
Chem. Phys. Lett.
(2000), 327, pp. 216. Link for the formation of exchange contributions. ⁸⁷⁹ Ochsenfeld C., Kussmann J., Koziol F.
Angew. Chem.
(2004), 116, pp. 4585. Link Since the derivative of the density matrix with respect to the magnetic field is skew-symmetric, its Coulomb-type contractions vanish. For the remaining Coulomb-type matrices the CFMM method ¹²⁷⁸ White C. A. et al.
Chem. Phys. Lett.
(1994), 230, pp. 8. Link is used. ⁸⁷⁹ Ochsenfeld C., Kussmann J., Koziol F.
Angew. Chem.
(2004), 116, pp. 4585. Link In addition, a multitude of different approaches for the solution of the CPSCF equations can be selected within Q-Chem.

To request a NMR chemical shift calculation using the density matrix approach, set JOBTYPE to NMR in the $rem section. Additional job-control variables can be found below.

D_CPSCF_PERTNUM

D_CPSCF_PERTNUM
       Specifies whether to do the perturbations one at a time, or all together.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Perturbed densities to be calculated all together. 1 Perturbed densities to be calculated one at a time.
RECOMMENDATION:
       None

D_SCF_CONV_1

D_SCF_CONV_1
       Sets the convergence criterion for the level-1 iterations. This preconditions the density for the level-2 calculation, and does not include any two-electron integrals.
TYPE:
       INTEGER
DEFAULT:
       4 corresponding to a threshold of $10^{-4}$ .
OPTIONS:
       $n<10$ Sets convergence threshold to $10^{-n}$ .
RECOMMENDATION:
       The criterion for level-1 convergence must be less than or equal to the level-2 criterion, otherwise the D-CPSCF will not converge.

D_SCF_CONV_2

D_SCF_CONV_2
       Sets the convergence criterion for the level-2 iterations.
TYPE:
       INTEGER
DEFAULT:
       4 Corresponding to a threshold of $10^{-4}$ .
OPTIONS:
       $n<10$ Sets convergence threshold to $10^{-n}$ .
RECOMMENDATION:
       None

D_SCF_MAX_1

D_SCF_MAX_1
       Sets the maximum number of level-1 iterations.
TYPE:
       INTEGER
DEFAULT:
       100
OPTIONS:
       $n$ User defined.
RECOMMENDATION:
       Use the default.

D_SCF_MAX_2

D_SCF_MAX_2
       Sets the maximum number of level-2 iterations.
TYPE:
       INTEGER
DEFAULT:
       30
OPTIONS:
       $n$ User defined.
RECOMMENDATION:
       Use the default.

D_SCF_DIIS

D_SCF_DIIS
       Specifies the number of matrices to use in the DIIS extrapolation in the D-CPSCF.
TYPE:
       INTEGER
DEFAULT:
       11
OPTIONS:
       $n$ $n$ = 0 specifies no DIIS extrapolation is to be used.
RECOMMENDATION:
       Use the default.

Example 10.41 NMR chemical shifts via the D-CPSCF method, showing all input options.

 $molecule
 0  1
 H        0.00000        0.00000        0.00000
 C        1.10000        0.00000        0.00000
 F        1.52324        1.22917        0.00000
 F        1.52324       -0.61459        1.06450
 F        1.52324       -0.61459       -1.06450
 $end

 $rem
    JOBTYPE            NMR
    EXCHANGE           B3LYP
    BASIS              6-31G*
    D_CPSCF_PERTNUM    0    D-CPSCF number of perturbations at once
    D_SCF_SOLVER       430  D-SCF   leqs_solver
    D_SCF_CONV_1       4    D-SCF   leqs_conv1
    D_SCF_CONV_2       4    D-SCF   leqs_conv2
    D_SCF_MAX_1        200  D-SCF   maxiter level 1
    D_SCF_MAX_2        50   D-SCF   maxiter level 2
    D_SCF_DIIS         11   D-SCF   DIIS
    D_SCF_ITOL         2    D-SCF   conv. criterion
 $end

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10.12.3 Linear-Scaling NMR Chemical Shift Calculations