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# 10.11.3 Linear-Scaling NMR Chemical Shift Calculations

(December 20, 2021)

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. 832 Ochsenfeld C., Kussmann J., Koziol F.
Angew. Chem.
(2004), 116, pp. 4585.
, 613 Kussmann J., Ochsenfeld C.
J. Chem. Phys.
(2007), 127, pp. 054103.
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. 835 Ochsenfeld C.
Phys. Chem. Chem. Phys.
(2000), 2, pp. 2153.
, 829 Ochsenfeld C. et al.
J. Am. Chem. Soc.
(2001), 123, pp. 2597.
, 137 Brown S. P. et al.
Angew. Chem. Int. Ed. Engl.
(2001), 40, pp. 717.
, 831 Ochsenfeld C. et al.
Solid State Nucl. Mag.
(2002), 22, pp. 128.
, 832 Ochsenfeld C., Kussmann J., Koziol F.
Angew. Chem.
(2004), 116, pp. 4585.
GIAO-HF/6-31G* calculations with 1,003 atoms and 8,593 basis functions, without symmetry, have been reported. 832 Ochsenfeld C., Kussmann J., Koziol F.
Angew. Chem.
(2004), 116, pp. 4585.
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 833 Ochsenfeld C., White C. A., Head-Gordon M.
J. Chem. Phys.
(1998), 109, pp. 1663.
, 834 Ochsenfeld C.
Chem. Phys. Lett.
(2000), 327, pp. 216.
for the formation of exchange contributions. 832 Ochsenfeld C., Kussmann J., Koziol F.
Angew. Chem.
(2004), 116, pp. 4585.
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 1210 White C. A. et al.
Chem. Phys. Lett.
(1994), 230, pp. 8.
is used. 832 Ochsenfeld C., Kussmann J., Koziol F.
Angew. Chem.
(2004), 116, pp. 4585.
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 10.38 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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