Slater-type orbital

Last updated January 27, 2025

Slater-type orbitals (STOs) or Slater-type functions (STFs) 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:

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
$\zeta$ 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

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

Hence

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 $Y_{l}^{m}(\mathbf {r} )$ depending on the polar coordinates of the position vector $\mathbf {r}$ as the angular part of the Slater orbital.

Derivatives

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

{\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

\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

\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

\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

\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

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

The Fourier transform is^[2]

{\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 $\omega$ are defined by

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

The overlap integral is

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

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

\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 ${\begin{aligned}\int \chi _{n\ell m}^{*}(\mathbf {r} ){\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}~\chi _{n'\ell 'm'}(\mathbf {r} ')~\mathrm {d} ^{3}r&=4\pi \int {\frac {1}{(2\pi )^{3}}}~\chi _{n\ell m}^{*}(\mathbf {k} )~{\frac {1}{k^{2}}}~\chi _{n'\ell 'm'}(\mathbf {k} )~\mathrm {d} ^{3}k\\&=8\,\delta _{\ell \ell '}\,\delta _{mm'}~(n-\ell )!~(n'-\ell )!~{\frac {\,(2\zeta )^{n}\,}{\zeta ^{\ell }}}{\frac {\,(2\zeta ')^{n'}\,}{\zeta '^{\ell }}}\int _{0}^{\infty }k^{2\ell }\left[\sum _{s=0}^{\lfloor (n-\ell )/2\rfloor }{\frac {\omega _{s}^{n\ell }}{(k^{2}+\zeta ^{2})^{n+1-s}}}\sum _{s'=0}^{\lfloor (n'-\ell )/2\rfloor }{\frac {\omega _{s'}^{n'\ell '}}{~~(k^{2}+\zeta '^{2})^{n'+1-s'}~}}\right]\mathrm {d} k\end{aligned}}$ 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]

Related Research Articles

In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. Due to the larger relative mass of a nucleus compared to an electron, the coordinates of the nuclei in a system are approximated as fixed, while the coordinates of the electrons are dynamic. The approach is named after Max Born and his 23-year-old graduate student J. Robert Oppenheimer, the latter of whom proposed it in 1927 during a period of intense ferment in the development of quantum mechanics.

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.

In computational chemistry and molecular physics, Gaussian orbitals are functions used as atomic orbitals in the LCAO method for the representation of electron orbitals in molecules and numerous properties that depend on these.

<span class="mw-page-title-main">Hurwitz zeta function</span> Special function in mathematics

In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables $s$ with $Re(s) > 1$ and $a \neq 0, -1, -2, \dots$ by

In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis. They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods.

In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons. Only a small subset of all possible fermionic wave functions can be written as a single Slater determinant, but those form an important and useful subset because of their simplicity.

In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude of torque the object experiences in a given magnetic field. When the same magnetic field is applied, objects with larger magnetic moments experience larger torques. The strength of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field. Its direction points from the south pole to north pole of the magnet.

In atomic physics, a term symbol is an abbreviated description of the total spin and orbital angular momentum quantum numbers of the electrons in a multi-electron atom. So while the word symbol suggests otherwise, it represents an actual value of a physical quantity.

The diabatic representation as a mathematical tool for theoretical calculations of atomic collisions and of molecular interactions.

The Artin reciprocity law, which was established by Emil Artin in a series of papers, is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.

In solid-state physics, the tight-binding model is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site. The method is closely related to the LCAO method used in chemistry. Tight-binding models are applied to a wide variety of solids. The model gives good qualitative results in many cases and can be combined with other models that give better results where the tight-binding model fails. Though the tight-binding model is a one-electron model, the model also provides a basis for more advanced calculations like the calculation of surface states and application to various kinds of many-body problem and quasiparticle calculations.

A normalized 1s Slater-type function is a function which is used in the descriptions of atoms and in a broader way in the description of atoms in molecules. It is particularly important as the accurate quantum theory description of the smallest free atom, hydrogen. It has the form

In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring $R$ , its representation space is generally denoted by $R (1)$ (that is, it is a representation $χ : G \to Aut R (R (1)) \approx GL(1, R)$ ).

<span class="mw-page-title-main">Helium atom</span> Atom of helium

A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with two neutrons, depending on the isotope, held together by the strong force. Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be used to estimate the ground state energy and wavefunction of the atom. Historically, the first such helium spectrum calculation was done by Albrecht Unsöld in 1927. Its success was considered to be one of the earliest signs of validity of Schrödinger's wave mechanics.

In cryptography, learning with errors (LWE) is a mathematical problem that is widely used to create secure encryption algorithms. It is based on the idea of representing secret information as a set of equations with errors. In other words, LWE is a way to hide the value of a secret by introducing noise to it. In more technical terms, it refers to the computational problem of inferring a linear $-ary function over a finite ring from given samples some of which may be erroneous. The LWE problem is conjectured to be hard to solve, and thus to be useful in cryptography.$

In probability theory and statistics, the generalized chi-squared distribution is the distribution of a quadratic form of a multinormal variable, or a linear combination of different normal variables and squares of normal variables. Equivalently, it is also a linear sum of independent noncentral chi-square variables and a normal variable. There are several other such generalizations for which the same term is sometimes used; some of them are special cases of the family discussed here, for example the gamma distribution.

In quantum mechanics, orbital magnetization, M_orb, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, M_spin, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non-magnetic material by an applied magnetic field.

In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions.

In condensed matter and atomic physics, Van Vleck paramagnetism refers to a positive and temperature-independent contribution to the magnetic susceptibility of a material, derived from second order corrections to the Zeeman interaction. The quantum mechanical theory was developed by John Hasbrouck Van Vleck between the 1920s and the 1930s to explain the magnetic response of gaseous nitric oxide and of rare-earth salts. Alongside other magnetic effects like Paul Langevin's formulas for paramagnetism and diamagnetism, Van Vleck discovered an additional paramagnetic contribution of the same order as Langevin's diamagnetism. Van Vleck contribution is usually important for systems with one electron short of being half filled and this contribution vanishes for elements with closed shells.

Non-linear inverse Compton scattering (NICS), also known as non-linear Compton scattering and multiphoton Compton scattering, is the scattering of multiple low-energy photons, given by an intense electromagnetic field, in a high-energy photon during the interaction with a charged particle, in many cases an electron. This process is an inverted variant of Compton scattering since, contrary to it, the charged particle transfers its energy to the outgoing high-energy photon instead of receiving energy from an incoming high-energy photon. Furthermore, differently from Compton scattering, this process is explicitly non-linear because the conditions for multiphoton absorption by the charged particle are reached in the presence of a very intense electromagnetic field, for example, the one produced by high-intensity lasers.

References

↑ Slater, J. C. (1930). "Atomic Shielding Constants". Physical Review . 36 (1): 57. Bibcode:1930PhRv...36...57S. doi:10.1103/PhysRev.36.57.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
Tai, H. (1986). "Analytic evaluation of two-center molecular integrals". Physical Review A . 33 (6): 3657–3666. Bibcode:1986PhRvA..33.3657T. doi:10.1103/PhysRevA.33.3657. PMID 9897107.
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.
Grotendorst, J.; Steinborn, E. O. (1988). "Numerical evaluation of molecular one- and two-electron multicenter integrals with exponential-type orbitals via the Fourier-transform method". Physical Review A . 38 (8): 3857–3876. Bibcode:1988PhRvA..38.3857G. doi:10.1103/PhysRevA.38.3857. PMID 9900838.
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.
Guseinov, I. I. (2001). "Evaluation of expansion coefficients for translation of Slater-Type orbitals using complete orthonormal sets of Exponential-Type functions". International Journal of Quantum Chemistry . 81 (2): 126–129. doi:10.1002/1097-461X(2001)81:2<126::AID-QUA2>3.0.CO;2-K.
Guseinov, I. I.; Mamedov, B. A. (2002). "On the calculation of arbitrary multielectron molecular integrals over Slater-Type Orbitals using recurrence relations for overlap integrals: III. auxiliary functions Q¹_nn' and G^q_−nn". International Journal of Quantum Chemistry . 86 (5): 440–449. doi:10.1002/qua.10045.
Guseinov, I. I.; Mamedov, B. A. (2002). "On the calculation of arbitrary multielectron molecular integrals over Slater-Type Orbitals using recurrence relations for overlap integrals: IV. Use of recurrence relations for basic two-center overlap and hybrid integrals". International Journal of Quantum Chemistry . 86 (5): 450–455. doi:10.1002/qua.10044.
Özdogan, T.; Orbay, M. (2002). "Evaluation of two-center overlap and nuclear attraction integrals over Slater-type orbitals with integer and non-integer principal quantum numbers". International Journal of Quantum Chemistry . 87 (1): 15–22. doi:10.1002/qua.10052.
Harris, F. E. (2003). "Comment on Computation of Two-Center Coulomb integrals over Slater-Type orbitals using elliptical coordinates". International Journal of Quantum Chemistry . 93 (5): 332–334. doi: 10.1002/qua.10567 .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[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.

[1]

[2]

[3]

[4]

[5]

[6]