Spin‐rotation constants and rotational g tensors can be evaluated as second derivatives of the energy with respect to the rotational angular momentum and nuclear spin or angular momentum and external magnetic field, respectively. To overcome problems with the slow basis set convergence and the unphysical (gauge‐)origin dependence in quantum chemical calculations of these two properties, we suggest the use of perturbation dependent atomic orbitals (rotational London orbitals), which depend explicitly on the angular momentum and the external magnetic field and are a generalization of the conventional London orbitals (also known as gauge‐including atomic orbitals). It is shown that calculations of spin‐rotation constants and rotational g tensors based on rotational London orbitals are closely related to London‐orbital computations of nuclear shieldings and magnetizabilities. Test calculations at the Hartree–Fock self‐consistent‐field level for HF, N2, CO, and CH2O demonstrate the superior convergence to the basis set limit provided by the rotational London orbitals. They suggest that future calculations employing rotational London orbitals in conjunction with highly correlated wave functions will be able to provide results of unprecedented accuracy for spin‐rotation constants and rotational g tensors.
REFERENCES
1.
W. H. Flygare, Molecular Structure and Dynamics (Prentice-Hall, Englewood Cliffs, 1978), Chap. 6.9;
2.
3.
4.
S. P. A.
Sauer
, V.
Spirko
, and J.
Oddershede
, Chem. Phys.
153
, 189
(1991
);S. P. A.
Sauer
, J.
Oddershede
, and J.
Geertsen
, Mol. Phys.
76
, 445
(1992
);5.
6.
7.
8.
K.
Wolinski
, J. F.
Hinton
, and P.
Pulay
, J. Am. Chem. Soc.
112
, 8251
(1990
).9.
10.
11.
M.
Häser
, R.
Ahlrichs
, H. P.
Baron
, P.
Weis
, and H.
Horn
, Theor. Chim. Acta
83
, 455
(1992
).12.
A suitable partitioning in case of perturbation-dependent basis functions is, for example, provided by the “natural connection,“ (Reference 18), but no improvement in basis set convergence of the dia-and paramagnetic terms is observed compared to results from conventional calculations [K. Ruud and T. Helgaker (unpublished)].
13.
Since the vector potential given in Eq. (8) for an external field is only correct if an additional scalar potential is introduced to ensure that the corresponding electric field vanishes everywhere. This scalar potential leads to the second-order contribution in the Hamiltonian (cf. Eq. (10)) which is required for the computation of rotational g tensors. Note also that only the diagonal elements of g in the pincipal axes system are well defined and contribute to the rotational Zeeman effect (see, for example, Ref. 1). In case of the spin-rotation constant, the choice of the vector potential due to the nuclear magnetic moments is only valid for coordinate systems fixed at the Kth nucleus. Transformation to the rotating coordinate system used to describe the electronic structure of the molecule yields an additional scalar potential which leads to the second-order contribution in the Hamiltonian (cf. Eq. (11)). Note that unlike most previous work we ignore all effects due to Thomas precession, as they can be shown to contribute only in higher orders.
14.
Analytic derivative methods have been the subject of several reviews. Examples include:
H. F.
Schaefer
and Y.
Yamaguchi
, J. Mol. Struct. (THEOCHEM)
135
, 369
(1986
);P. Pulay, in Advances in Chemical Physics, Ab Initio Methods in Quantum Chemistry II, edited by K. P. Lawley (Wiley, New York, 1987), pp. 241–286;
J.
Gauss
and D.
Cremer
, Adv. Quantum. Chem.
23
, 205
(1992
)., Adv. Quantum Chem.
15.
Related, so-called velocity-gauge basis sets have been suggested by
L. A.
Nafie
[J. Chem. Phys.
96
, 5687
(1992
)] within the theory of vibrational circular dichroism.16.
See, for example, L. D. Landau and E. M. Lifschitz, Lehrbuch der Theoretischen Physik, Bd II Feldtheorie (Akademie-Verlag, Berlin, 1981), Chap. V.
17.
For completeness, we note that origin-independent calculations of spinrotation constants based on Eq. (34) have been already presented in the literature:
R.
Ditchfield
, J. Chem. Phys.
56
, 5688
(1972
);D.
Sundholm
, J.
Gauss
, and R.
Ahlrichs
, Chem. Phys. Lett.
243
, 264
(1995
), though to the best of our knowledge a theoretical justification for the applied procedure has never been given. 18.
J.
Olsen
, K. L.
Bak
, K.
Ruud
, T.
Helgaker
, and P.
Jo/rgensen
, Theor. Chim. Acta
90
, 421
(1995
).19.
However, we note that for properties that cannot be expressed as energy derivatives, like for instance electronic circular dichroism (ECD), which is a residue of the linear response function:
K. L.
Bak
, Aa. E.
Hansen
, K.
Ruud
, T.
Helgaker
, J.
Olsen
, and P.
Jo/rgensen
, Theor. Chim. Acta
90
, 441
(1995
), the natural connection provides the only suitable definition of connection for perturbation dependent basis set.20.
K. Ruud, P.-O. Åstrand, T. Helgaker, and K. V. Mikkelsen, J. Mol. Struct. ( THEOCHEM) ( in press);
J. Gauss and D. Sundholm ( to be published).
21.
ACES II, an ab initio program system, authored by J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. Bartlett. The package also contains modified versions of the MOLECULE Gaussian integral program of J. Almlöf and P. R. Taylor, the ABACUS integral derivative program written by T. Helgaker, H. J. Aa. Jensen, P. Jo/rgensen and P. R. Taylor, and the PROPS property integral code of P. R. Taylor. For a detailed description of ACES II, see, also
J. F.
Stanton
, J.
Gauss
, J. D.
Watts
, W. J.
Lauderdale
, and R. J.
Bartlett
, Int. J. Quantum Chem. Symp.
26
, 879
(1992
).22.
T. Helgaker, H. J. Aa. Jensen, P.Jo/rgensen, H. Koch, J. Olsen, H. Ågren, K. L. Bak, V. Bakken, O. Christiansen, P. Dahle, E. K. Dalskov, T. Enevoldsen, A. Halkier, H. Heiberg, D. Jonsson, S. Kirpekar, R. Kobayashi, A. S. de Meras, K. V. Mikkelsen, P. Norman, M. J. Packer, K. Ruud, P. R. Taylor, and O. Vahtras: DALTON QCP—an electronic structure program.
23.
The used geometries are for for for for formaldehyde: 〈(HCO) = 121.88°.
24.
A.
Schäfer
, H.
Horn
, and R.
Ahlrichs
, J. Chem. Phys.
97
, 2571
(1992
).25.
dzp: ( 8s4p1d/4s2p1d) for first-row elements and ( 4s1p/2s1p) for H;
tzp: ( 9s5p1d/5s3p1d) and ( 5s1p/3s1p);
qz2d1f: ( 11s7p2d1f/6s4p2d1f) and ( 6s2p1d/3s2p1d);
pz3d2f: ( 13s8p3d2f/8s5p3d2f) and ( 8s3p2d/6s3p2d). All calculations were carried out with spherical polarization functions.
26.
D. Sundholm, J. Gauss and A. Schäfer, J. Chem. Phys. (submitted).
27.
See, for example, Ref. 10.
28.
K.
Ruud
, T.
Helgaker
, P.
Jo.rgensen
, and K. L.
Bak
, Chem. Phys. Lett.
223
, 12
(1994
).29.
K.
Ruud
, H.
Skaane
, T.
Helgaker
, K. L.
Bak
, and P.
Jo.rgensen
, J. Am. Chem. Soc.
116
, 10135
(1994
).30.
31.
R. A.
Kendall
, T. H.
Dunning
Jr., and R. J.
Harrison
, J. Chem. Phys.
96
, 6796
(1992
).32.
33.
One might wonder why we have chosen different basis sets for the calculation of spin-rotation constants and rotational g tensors. However, previous calculations have shown that Dunning’s correlation consistent basis sets are not well suited for computations of nuclear shieldings (probably due to a too tight contraction of the core orbitals), while test calculations indicated that the “Karlsruhe“ basis sets are not optimal for magnetizabilities.
34.
35.
P. Dahle, K. Ruud, T. Helgaker, K. L. Bak and P. Jo/rgensen (unpublished).
This content is only available via PDF.
© 1996 American Institute of Physics.
1996
American Institute of Physics
You do not currently have access to this content.
