Currently, I’m working on adding nuclear magnetic resonance (NMR) chemical shift calculations to GAMESS for closed-shell molecules. If you’ve been through sophomore organic chemistry, you know that NMR is an extremely useful spectroscopic tool. You also know that it can be difficult to assign certain shifts to certain atoms in the spectrum; this is where theory can help sort out experimental observations. NMR shift calculations have been included in quantum chemistry packages for many years, but our twist on the idea will be to couple the NMR formalism with our effective fragment potential method, which will (hopefully) allow for the calculation of chemical shifts in solution.
The NMR chemical shift formalism was worked out by Robert Ditchfield in 1974 (Mol. Phys., 27, 789 (1974)). The main problem with calculating NMR shifts at the time was something called gauge dependence: As you know, NMR spectra are obtained by placing the sample to be tested in an external magnetic field. Mathematically, this field is represented by a potential, which depends on the choice of a specific origin in space. The way the potential is defined, you could pick any origin and still reproduce the applied magnetic field.
In Hartree-Fock theory, this is fine as long as you’re using a complete (infinite) basis. …What’s that? Can’t find the option to use an infinite basis set in your favorite quantum chemistry program? Then you’re in a little fix — it turns out that the chemical shift you calculate with a finite basis will depend on your choice of origin for the magnetic potential. Not good.
Ditchfield solved this problem by making the basis set itself magnetic field-dependent. He called these functions Gauge-Invariant Atomic Orbitals, since they lead to a gauge-invariant shielding. Pople pointed out that this probably isn’t the best name, since the functions depend on the field, so now you still see GIAO used, but the "I" stands for everything from "including" to "independent," depending on which paper you read. (I think "invariant" is fine.)
Anyway, at the end of the day, you get the following expression for NMR chemical shifts, in the GIAO basis, at the RHF level of theory:
In the above equation, sigma is the chemical shift; the subscript B refers to any of the nuclei in the molecule, and alpha and beta are each of the three Cartesian directions x, y, or z. The sum is over all field independent basis functions; the field dependence comes in through the T and Q matrices, and these depend on the nuclear positions. L is the one-electron spin-orbit operator, but the details of that aren’t critical for our purposes here.
At this point, I’ve got code written that does all the one-electron integrals seen above. These are done using the REALLY spiffy McMurchie-Davidson integral scheme (J. Comp. Phys. 26, 218, (1978)). The guts of this routine were taken from NWChem. (Thanks Ricky.)
For H2, the one-electron integrals
in the second term of the shielding tensor are all zero, so I can calculate
the full chemical shift for that molecule. I get, with 10 basis functions
NUCLEAR MAGNETIC RESONANCE: RHF CHEMICAL SHIFTS
FIRST TERM OF SHIELDING TENSOR (ppm):
| X | Y | Z | ||
| 1 H1 | X | 26.4351 | 0 | 0 |
| Y | 0 | 26.4351 | 0 | |
| Z | 0 | 0 | 27.7230 |
which averages to 26.8644 ppm. One published theoretical result with 100 basis functions is 26.8139 ppm, and an experimental result is 26.396(4) ppm. I haven’t done extensive testing yet (since I’m not done with the code) but this seems promising.
I’m working now on the two-electron integrals
I need for the derivative of the density matrix. These are easy to evaluate
in the McMurchie-Davison scheme, but it’s a bit harder to get the integrals
in the format that GAMESS packs them in. Once this is done, I’ll write
some Coupled-Perturbed Hartree Fock code to get the full derivative of
the density matrix, and the RHF NMR shifts will be done. Then it’s time
to couple the formalism with fragments…