Slater-type orbital

Slater-type orbitals (STOs) are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method. They are named after the physicist John C. Slater, who introduced them in 1930.^[1]

They possess exponential decay at long range and Kato's cusp condition at short range (when combined as hydrogen-like atom functions, i.e. the analytical solutions of the stationary Schrödinger equation for one electron atoms). Unlike the hydrogen-like ("hydrogenic") Schrödinger orbitals, STOs have no radial nodes (neither do Gaussian-type orbitals).

Definition

STOs have the following radial part:

${\displaystyle R(r)=Nr^{n-1}e^{-\zeta r}\,}$

where

• n is a natural number that plays the role of principal quantum number, n = 1,2,...,
• N is a normalizing constant,
• r is the distance of the electron from the atomic nucleus, and
• ${\displaystyle \zeta }$ is a constant related to the effective charge of the nucleus, the nuclear charge being partly shielded by electrons. Historically, the effective nuclear charge was estimated by Slater's rules.

The normalization constant is computed from the integral

${\displaystyle \int _{0}^{\infty }x^{n}e^{-\alpha x}\,\mathrm {d} x={\frac {n!}{~\alpha ^{n+1}\,}}~.}$

Hence

${\displaystyle N^{2}\int _{0}^{\infty }\left(r^{n-1}e^{-\zeta r}\right)^{2}r^{2}\,\mathrm {d} r=1\Longrightarrow N=(2\zeta )^{n}{\sqrt {\frac {2\zeta }{(2n)!}}}~.}$

It is common to use the spherical harmonics ${\displaystyle Y_{l}^{m}(\mathbf {r} )}$ depending on the polar coordinates of the position vector ${\displaystyle \mathbf {r} }$ as the angular part of the Slater orbital.

Derivatives

The first radial derivative of the radial part of a Slater-type orbital is

${\displaystyle {\partial R(r) \over \partial r}=\left[{\frac {(n-1)}{r}}-\zeta \right]R(r)}$

The radial Laplace operator is split in two differential operators

${\displaystyle \nabla ^{2}={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial \over \partial r}\right)}$

The first differential operator of the Laplace operator yields

${\displaystyle \left(r^{2}{\partial \over \partial r}\right)R(r)=\left[(n-1)r-\zeta r^{2}\right]R(r)}$

The total Laplace operator yields after applying the second differential operator

${\displaystyle \nabla ^{2}R(r)=\left({1 \over r^{2}}{\partial \over \partial r}\right)\left[(n-1)r-\zeta r^{2}\right]R(r)}$

the result

${\displaystyle \nabla ^{2}R(r)=\left[{n(n-1) \over r^{2}}-{2n\zeta \over r}+\zeta ^{2}\right]R(r)}$

Angular dependent derivatives of the spherical harmonics don't depend on the radial function and have to be evaluated separately.

Integrals

The fundamental mathematical properties are those associated with the kinetic energy, nuclear attraction and Coulomb repulsion integrals for placement of the orbital at the center of a single nucleus. Dropping the normalization factor N, the representation of the orbitals below is

${\displaystyle \chi _{n\ell m}({\mathbf {r} })=r^{n-1}~e^{-\zeta \,r}~Y_{\ell }^{m}({\mathbf {r} })~.}$

The Fourier transform is^[2]

${\displaystyle {\begin{aligned}\chi _{n\ell m}({\mathbf {k} })&=\int e^{i{\mathbf {k} }\cdot {\mathbf {r} }}~\chi _{n\ell m}({\mathbf {r} })~\mathrm {d} ^{3}r\\&=4\pi ~(n-\ell )!~(2\zeta )^{n}~(ik/\zeta )^{\ell }~Y_{\ell }^{m}({\mathbf {k} })\sum _{s=0}^{\lfloor (n-\ell )/2\rfloor }{\frac {\omega _{s}^{n\ell }}{~(k^{2}+\zeta ^{2})^{n+1-s}}},\end{aligned}}}$

where the ${\displaystyle \omega }$ are defined by

${\displaystyle \omega _{s}^{n\ell }\equiv \left(-{\frac {1}{4\zeta ^{2}}}\right)^{s}\,{\frac {(n-s)!}{~s!~(n-\ell -2s)!~}}.}$

The overlap integral is

${\displaystyle \int \chi _{n\ell m}^{*}(r)~\chi _{n'\ell 'm'}(r)~\mathrm {d} ^{3}r=\delta _{\ell \ell '}\,\delta _{mm'}\,{\frac {(n+n')!}{~(\zeta +\zeta ')^{n+n'+1}}}}$

of which the normalization integral is a special case. The superscript star denotes complex-conjugation.

The kinetic energy integral is $${\displaystyle {\begin{aligned}&\int \chi _{n\ell m}^{*}(r)~\left(-{\tfrac {1}{2}}\nabla ^{2}\right)\,\chi _{n'\ell 'm'}(r)~\mathrm {d} ^{3}r\\&={\frac {1}{2}}\delta _{\ell \ell '}\,\delta _{mm'}\,\int _{0}^{\infty }e^{-(\zeta +\zeta ')\,r}\left[[\ell '(\ell '+1)-n'(n'-1)]\,r^{n+n'-2}+2\zeta 'n'\,r^{n+n'-1}-\zeta '^{2}\,r^{n+n'}\right]~\mathrm {d} r~,\end{aligned}}}$$ a sum over three overlap integrals already computed above.

The Coulomb repulsion integral can be evaluated using the Fourier representation (see above)

${\displaystyle \chi _{n\ell m}^{*}({\mathbf {r} })=\int {\frac {~e^{i\mathbf {k} \cdot \mathbf {r} }~}{(2\pi )^{3}}}~\chi _{n\ell m}^{*}({\mathbf {k} })~\mathrm {d} ^{3}k}$

which yields $${\displaystyle {\begin{aligned}\int \chi _{n\ell m}^{*}(\mathbf {r} ){\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}~\chi _{n'\ell 'm'}(\mathbf {r} ')~\mathrm {d} ^{3}r&=4\pi \int {\frac {1}{(2\pi )^{3}}}~\chi _{n\ell m}^{*}(\mathbf {k} )~{\frac {1}{k^{2}}}~\chi _{n'\ell 'm'}(\mathbf {k} )~\mathrm {d} ^{3}k\\&=8\,\delta _{\ell \ell '}\,\delta _{mm'}~(n-\ell )!~(n'-\ell )!~{\frac {\,(2\zeta )^{n}\,}{\zeta ^{\ell }}}{\frac {\,(2\zeta ')^{n'}\,}{\zeta '^{\ell }}}\int _{0}^{\infty }k^{2\ell }\left[\sum _{s=0}^{\lfloor (n-\ell )/2\rfloor }{\frac {\omega _{s}^{n\ell }}{(k^{2}+\zeta ^{2})^{n+1-s}}}\sum _{s'=0}^{\lfloor (n'-\ell )/2\rfloor }{\frac {\omega _{s'}^{n'\ell '}}{~~(k^{2}+\zeta '^{2})^{n'+1-s'}~}}\right]\mathrm {d} k\end{aligned}}}$$ These are either individually calculated with the law of residues or recursively as proposed by Cruz et al. (1978).^[3]

STO software

Some quantum chemistry software uses sets of Slater-type functions (STF) analogous to Slater type orbitals, but with variable exponents chosen to minimize the total molecular energy (rather than by Slater's rules as above). The fact that products of two STOs on distinct atoms are more difficult to express than those of Gaussian functions (which give a displaced Gaussian) has led many to expand them in terms of Gaussians.^[4]

Analytical ab initio software for polyatomic molecules has been developed, e.g., STOP: a Slater Type Orbital Package in 1996.^[5]

SMILES uses analytical expressions when available and Gaussian expansions otherwise. It was first released in 2000.

Various grid integration schemes have been developed, sometimes after analytical work for quadrature (Scrocco), most famously in the ADF suite of DFT codes.

After the work of John Pople, Warren. J. Hehre and Robert F. Stewart, a least squares representation of the Slater atomic orbitals as a sum of Gaussian-type orbitals is used. In their 1969 paper, the fundamentals of this principle are discussed and then further improved and used in the GAUSSIAN DFT code. ^[6]

See also

• Basis sets used in computational chemistry

References

1. Slater, J. C. (1930). "Atomic Shielding Constants". Physical Review. 36 (1): 57. Bibcode:1930PhRv...36...57S. doi:10.1103/PhysRev.36.57.
2. Belkic, D.; Taylor, H. S. (1989). "A unified formula for the Fourier transform of Slater-type orbitals". Physica Scripta. 39 (2): 226–229. Bibcode:1989PhyS...39..226B. doi:10.1088/0031-8949/39/2/004. S2CID 250815940.
3. Cruz, S. A.; Cisneros, C.; Alvarez, I. (1978). "Individual orbit contribution to the electron stopping cross section in the low-velocity region". Physical Review A. 17 (1): 132–140. Bibcode:1978PhRvA..17..132C. doi:10.1103/PhysRevA.17.132.
4. Guseinov, I. I. (2002). "New complete orthonormal sets of exponential-type orbitals and their application to translation of Slater Orbitals". International Journal of Quantum Chemistry. 90 (1): 114–118. doi:10.1002/qua.927.
5. Bouferguene, A.; Fares, M.; Hoggan, P. E. (1996). "STOP: Slater Type Orbital Package for general molecular electronic structure calculations". International Journal of Quantum Chemistry. 57 (4): 801–810. doi:10.1002/(SICI)1097-461X(1996)57:4<801::AID-QUA27>3.0.CO;2-0.
6. Hehre, W. J.; Stewart, R. F.; Pople, J. A. (1969-09-15). "Self‐Consistent Molecular‐Orbital Methods. I. Use of Gaussian Expansions of Slater‐Type Atomic Orbitals". The Journal of Chemical Physics. 51 (6): 2657–2664. Bibcode:1969JChPh..51.2657H. doi:10.1063/1.1672392. ISSN 0021-9606.

• Harris, F. E.; Michels, H. H. (1966). "Multicenter integrals in quantum mechanics. 2. Evaluation of electron-repulsion integrals for Slater-type orbitals". Journal of Chemical Physics. 45 (1): 116. Bibcode:1966JChPh..45..116H. doi:10.1063/1.1727293.
• Filter, E.; Steinborn, E. O. (1978). "Extremely compact formulas for molecular two-center and one-electron integrals and Coulomb integrals over Slater-type atomic orbitals". Physical Review A. 18 (1): 1–11. Bibcode:1978PhRvA..18....1F. doi:10.1103/PhysRevA.18.1.
• McLean, A. D.; McLean, R. S. (1981). "Roothaan-Hartree-Fock Atomic Wave Functions, Slater Basis-Set Expansions for Z = 55–92". Atomic Data and Nuclear Data Tables. 26 (3–4): 197–381. Bibcode:1981ADNDT..26..197M. doi:10.1016/0092-640X(81)90012-7.
• Datta, S. (1985). "Evaluation of Coulomb integrals with hydrogenic and Slater-type orbitals". Journal of Physics B. 18 (5): 853–857. Bibcode:1985JPhB...18..853D. doi:10.1088/0022-3700/18/5/006.
• Grotendorst, J.; Steinborn, E. O. (1985). "The Fourier transform of a two-center product of exponential-type functions and its efficient evaluation". Journal of Computational Physics. 61 (2): 195–217. Bibcode:1985JCoPh..61..195G. doi:10.1016/0021-9991(85)90082-8.
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• Grotendorst, J.; Weniger, E. J.; Steinborn, E. O. (1986). "Efficient evaluation of infinite-series representations for overlap, two-center nuclear attraction, and Coulomb integrals using nonlinear convergence accelerators". Physical Review A. 33 (6): 3706–3726. Bibcode:1986PhRvA..33.3706G. doi:10.1103/PhysRevA.33.3706. PMID 9897112.
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• Bunge, C. F.; Barrientos, J. A.; Bunge, A. V. (1993). "Roothaan-Hartree-Fock Ground-State Atomic Wave Functions: Slater-Type Orbital Expansions and Expectation Values for Z=2–54". Atomic Data and Nuclear Data Tables. 53 (1): 113–162. Bibcode:1993ADNDT..53..113B. doi:10.1006/adnd.1993.1003.
• Harris, F. E. (1997). "Analytic evaluation of three-electron atomic integrals with Slater wave functions". Physical Review A. 55 (3): 1820–1831. Bibcode:1997PhRvA..55.1820H. doi:10.1103/PhysRevA.55.1820.
• Ema, I.; García de La Vega, J. M.; Miguel, B.; Dotterweich, J.; Meißner, H.; Steinborn, E. O. (1999). "Exponential-type basis functions: single- and double-zeta B function basis sets for the ground states of neutral atoms from Z=2 to Z=36". Atomic Data and Nuclear Data Tables. 72 (1): 57–99. Bibcode:1999ADNDT..72...57E. doi:10.1006/adnd.1999.0809.
• Fernández Rico, J.; Fernández, J. J.; Ema, I.; López, R.; Ramírez, G. (2001). "Four-center integrals for Gaussian and Exponential Functions". International Journal of Quantum Chemistry. 81 (1): 16–28. doi:10.1002/1097-461X(2001)81:1<16::AID-QUA5>3.0.CO;2-A.
• Guseinov, I. I.; Mamedov, B. A. (2001). "On the calculation of arbitrary multielectron molecular integrals over Slater-Type Orbitals using recurrence relations for overlap integrals: II. Two-center expansion method". International Journal of Quantum Chemistry. 81 (2): 117–125. doi:10.1002/1097-461X(2001)81:2<117::AID-QUA1>3.0.CO;2-L.
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