Clarkson University Relativistic Effective Potential Database

L.A. Fefee, S.A. Wildman, G.A. DiLabio, T.M. Moffett, Jr., J.C. Peploski and P.A. Christiansen

Department of Chemistry

Clarkson University

Potsdam, New York 13699-5810

Background

The relativistic effective potentials in this data base were generated by the shape consistent procedure proposed by Christiansen et al.¹ and subsequently generalized to include relativity² using the methodology developed by Lee, Ermler and Pitzer³ . Briefly, one first solves the Dirac-Hartree-Fock equations to obtain four-component atomic spinors. The four-component valence spinors, _lj, are then transformed to two components,

_lj = _lj + F_lj, (1)

where F_lj is itself a four-component spinor with large components that cancel the radial core-like oscillations in the large components of the _lj, but small components that completely cancel those of the _lj. The two-component pseudospinor, _lj, is then effectively reinserted into the DHF equation and the equation inverted to obtain the localized relativistic effective potential, U^REP_lj. The total REP could then be written as an infinite expansion of the U_lj^REP, each with the appropriate projection operators. In practice, of course, the expansion is truncated at the lowest angular momentum value, L, for which there are no longer any core-like oscillations in the large component of _LJ. That is, the large components of F_LJ disappear. Due to the minimal exchange interaction between the core and higher l spinors and the absence of the F_lj large components for all l greater than L, this truncation is an excellent approximation. For use in conventional codes, the REPs are then averaged and differenced for each l to form the corresponding averaged relativistic effective potentials (AREP) along with spin-orbit operators.

Averaged relativistic effective potentials with effective spin-orbit operators from Christiansen and coworkers² are tabulated here as Gaussian expansions,

U_l^{(AREP or SO)} = r^-2 C_lir^n_li exp(-_lir²), (2)

as originally proposed by Kahn⁴ and others. As such the AREP can be used without modification in standard software such as Gaussian-94⁵ or the Columbus program package⁶. As required for the Columbus package, the tabulated spin-orbit operator coefficients have been multiplied by factors of 2/l.

While the molecular small components are not treated explicitly in the shape consistent REP formalism, the small components, along with the core oscillations, are included in the effective potential through the pseudospinor transformation and subsequent Fock equation inversion. Indeed, implicit variation of the small components in subsequent molecular environments is allowed by means of the projection operators.

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Last updated: August 6, 1997

Errors

In addition to errors resulting from the frozen core approximation, errors arise from the effective potential approximation as well. Fortunately in the shape consistent formalism the key sources are fairly easy to identify and minimize. For a single valence electron and ignoring relativity we can write the atomic Fock equation as,

(-1/2 ² -Z_nuc/r + W_core) = (3)

where Z_nuc is the nuclear charge, W_core includes the coulomb and exchange interactions with the core, is the energy eigenvalue and is the valence eigenfunction. By partitioning along the lines of Eq. 1, the Fock equation becomes,

(¹/₂² +Z_nuc/r - W_core + )F + (-Z_core/r + W_core) + (-¹/₂² -1/r) = , (4)

and the corresponding effective potential, U^EP, is then,

U^EP = {(¹/₂² +Z_nuc/r - W_core + )F + (-Z_core/r + W_core)}/. (5)

The last term on the right, {(-Z_core/r + W_core)}/, is just the localized interaction potential between the frozen core and the valence electron and, except for exchange in W_core, is exact within the frozen core approximation. The exchange terms fortunately include the core orbitals as multiplicative factors and as a consequence their contributions in the valence space will decay roughly according to the core/valence amplitude ratios. The other term, {(¹/₂² +Z_nuc/r - W_core + )F}/, however is entirely the result of the pseudoorbital transformation. This is the strong repulsive term that prevents callapse into the core. This term will disappear in the valence region where and are identical, but in the inner valence (or outer core) region it may cause serious problems. Clearly to ensure transferability one must choose the core space to be small enough to minimize the local exchange approximation in the valence or bonding regions, and for a given core space, one must form the pseudoorbitals such that the F go to zero as close to the origin as possible.

It has recently been shown that effective potential bond length errors observed for the sixth-row main group elements were the result of F function tails in the generation of the 5f pseudospinors. As a consequence the sixth-row main group REPs included in this data base are from S.A. Wildman, et al.^2f As seen in the table below, the new REPs give hydride bond lengths in excellent agreement with experimental values.

Computed hydride bond lengths from new SPD type REPs used in selected intermediate coupling CI (SICCI) calculations. Experimental values from ref. 7 are included for comparison.

Method                          TlH                      PbH                    BiH


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SICCI d-shell occupation


	10(frozen d)		1.907			1.861			1.818


	9			1.843			1.833			1.791


	8			1.874			1.852			1.795





Experiment		        1.870			1.839			1.805




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Note the importance shown in the table of d-shell correlation (d⁸ occupation). From the above results it is absolutely essential that the 5d shell be included in the valence space for the sixth-row main group elements.

Errors will also arise from the Gaussian expansions of the AREPs and SO operators, but these are generally fairly small. The AREPs and SO operators should not be truncated. Note however that the atomic orbital basis sets provided for use with the REPs are of minimal size and should generally be supplemented with diffuse and polarization functions where higher accuracy is required.

References

P.A. Christiansen, Y.S. Lee and K.S. Pitzer, J. Chem. Phys. 71, 4445 (1979).

(a) L.F. Pacios and P.A. Christiansen, J. Chem. Phys. 82, 2664 (1985); (b) M.M. Hurley, L.F. Pacios, P.A. Christiansen, R.B. Ross and W.C. Ermler, J. Chem. Phys. 84, 6840 (1986); (c) L.A. LaJohn, P.A. Christiansen, R.B. Ross, T. Atashroo and W.C. Ermler, J. Chem. Phys. 87, 2812 (1987); (d) R.B. Ross, J.M. Powers, T. Atashroo, W.C. Ermler, L.A. LaJohn and P.A. Christiansen, J. Chem. Phys. 93, 6654 (1990); (e)R.B. Ross, J.M. Powers, T. Atashroo, W.C. Ermler, L.A. LaJohn and P.A. Christiansen, Erratum, J. Chem. Phys. 101, 1098 (1994); (f) S.A. Wildman, G.A. DiLabio and P.A. Christiansen, J. Chem. Phys. (Submitted).

Y.S. Lee, W.C. Ermler and K.S. Pitzer, J. Chem. Phys. 67, 5861 (1977).

L.R. Kahn, P. Baybutt and D.G. Truhlar, J. Chem. Phys. 65, 3826 (1976).

M.J. Frisch, G.W. Trucks, M. Head-Gordon, P.M.W. Schlegel, M.A. Robb, E.S. Repogle, R. Gomperts, J.L. Andres, K. Raghavachari, J.S. Binkley, C. Gonzales, R.L. Martin, D.J. Fox, D.J. Defrees, J. Baker, J.J.P. Stewart and J. A. Pople, GAUSSIAN-94, Gaussian Inc., Pittsburgh, PA (1992).

A.A.H. Chang and R.M. Pitzer, J. Am. Chem. Soc. 111, 2500 (1989), and references therein.

K.P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV, Constants of Diatomic Molecules, (Van Nostrand Reinhold, 1979).