Hartmann potential

The Hartmann potential in theoretical chemistry is a ring-shaped potential field, which, in spherical coordinates, is a function of the ring radius and the polar angle : ${\ displaystyle V}$ ${\ displaystyle r}$ ${\ displaystyle \ theta}$

{\ displaystyle V (r, \ theta) = \ left ({\ frac {2r_ {0}} {r}} - {\ frac {r_ {0} ^ {2}} {r ^ {2} \ sin ^ {2} \ theta}} \ right) V_ {0}}

The minimum of the potential well and the radial distance ,, of the potential minimum from the center of the potential ring are two system-specific quantities. The movement of a body under the influence of the Hartmann potential can be specified exactly and in closed form. ${\ displaystyle V_ {0}}$ ${\ displaystyle r_ {0}}$

Importance to chemistry

The potential energy published by Hermann Hartmann for the first time in 1972 was the result of his considerations, begun in 1940, to understand more precisely ring-shaped molecules such as benzene and thus also the Hückel model of aromatic compounds and the HMO method (Hückel molecular orbital, see Hückel approximation ) developed from it .

For the first time, the Hartmann student Karl Jug formulated a ring-shaped potential energy in his dissertation (1965), which allows the spectrum as well as electron density, dipole moment, transition moments, excitation energies and diamagnetism of heterocyclic systems to be understood semi-quantitatively. To this end, K. Jug transferred a single-center approach presented by H. Hartmann in 1947 using the example of the CH ₄ molecule to ring-shaped molecules such as furan and pyrrole (five-heterocycles). In the following years, other Hartmann employees established the importance of ring-shaped potentials using benzene as an example (KH Hansen and E. Frenkel, 1966, and H. von Hirschhausen, 1970). All of these chemically important findings were integrated into his ring-shaped potential field by H. Hartmann.

Starting with the investigations by Maurice Kibler, Lyon, who coined the term "Hartmann potential", the general significance of this potential for the theory of chemical bonds became clear in the 1980s. Due to the self-coupling on which the Hartmann potential is based, possibilities which are applied in the hydrogen atom but are still undifferentiated are specified. The singular central symmetry of the hydrogen atom unfolds through the Hartmann potential to the axial bond dynamics, which makes previously hidden behavior patterns visible.

Axial non-central valence fields

The Hartmann potential extends the three-dimensional Kepler-Coulomb potential of the hydrogen atom by a 1 / r² term, which corresponds to a self-coupling of the radial degree of freedom. Due to this non-linear addition, the central order of the hydrogen atom takes a back seat and an axial non-central potential arises, as is typical for bonding systems. Axiality of the valence field is found in the entire area of the chemical bond, not only in molecules such as benzene, but also in the two-center bond but also in large molecules such as DNA .

Hydrogen atom as a reference system

Because the symmetry of the hydrogen atom is the hidden symmetry of the Hartmann potential, the two system characteristic variables can the potential on the Bohr radius of hydrogen , or the ground state energy of the hydrogen atom are related. This is achieved by the Hartmann parameters and , two positive numbers. ${\ displaystyle a_ {0}}$ ${\ displaystyle E_ {0}}$ ${\ displaystyle \ eta}$ ${\ displaystyle \ sigma}$

{\ displaystyle r_ {0} = \ eta a_ {0}}

{\ displaystyle V_ {0} = \ sigma ^ {2} E_ {0}}

The Hartmann potential therefore makes the hydrogen atom a direct reference system for axial bonding systems. The fundamental importance of the Hartmann potential is based on this. The inherent dynamics valid for the hydrogen atom unfold through the Hartmann potential for bonding dynamics.

Because of its importance for theoretical chemistry, the Hartmann potential has been the subject of numerous studies. The main findings concern

the relationship to other potentials,
the exact solvability and
the underlying dynamic symmetry.

Relationship to other potentials

Ring-shaped potentials: From a mathematical point of view, the Hartmann potential belongs to the family of ring-shaped potentials, which are derived from the Kepler-Coulomb potential. These potentials also include the scratch potential (1920), one of the four Smorodinski-Winternitz potentials (1967) and general MIC-Kepler systems (1968/1970). Like the Hartmann potential, all of these potentials can be solved exactly. Annular potentials describe non-central valence fields that are located around an axis.
Duality to the ring-shaped oscillator: The duality relationship already existing between the Kepler-Coulomb potential and the isotropic oscillator also applies between the family of ring-shaped Kepler-Coulomb potentials and the family of ring-shaped oscillators,
Unification of all ring-shaped systems: The two families of ring-shaped potentials can be combined into one family by regularization. This enables a uniform treatment of all ring-shaped potentials. The Kustaanheimo – Stiefel transformation or spinor regularization transforms the non-linear and singular equations of motion into linear and regular equations of harmonic 4D oscillators. The background to this regularization is the consideration of spatial orientation.
Perturbation calculation for N-body systems: Four-dimensional harmonic oscillators are the starting point for the systematic application of perturbation theory when calculating the behavior of N-body systems. E.g. for the analytical calculation of the classical trajectories when determining the transition dipole moments or excitation energies of molecular interactions within the framework of the semi-classical S-matrix.

Exact solvability

As for the Kepler-Coulomb potential of the hydrogen atom, the energy eigenvalues and the associated wave functions of the radial Schrödinger equation can also be specified exactly and in closed form for the movement of a body or particle in the Hartmann potential field . According to H. Hartmann's calculations, the eigenvalues for the ring-shaped potential are always proportional to the ground state energy of the hydrogen atom. For the ground state of the ring-shaped potential, the proportionality factor is determined solely by the Hartmann parameters, while the quantum numbers of the hydrogen atom are also precisely taken into account for the excited bound states . In this way, the Hartmann potential makes the hydrogen atom the reference system for axial bonding systems.

Quantum chemistry of the models

According to H. Hartmann, precisely solvable potentials are important for two reasons:

The phenomena that are important for fundamental reasons, such as the transition dipole moment and the excitation function, can be studied quantitatively on exactly solvable potentials. If the solution is in a closed form, you not only have numerical values, but also manageable relationships.
In the quantum mechanical consideration of complicated atomic and molecular systems, typical model systems that relate to exactly solvable potentials play a role for fundamental reasons.

Super integrability

According to Noether's theorem ( Emmy Noether , 1918), investigations into the solvability of classical and quantum mechanical equations of motion bring to light the properties of the underlying dynamic symmetry. Exact solvability always means that there are more motion integrals (conservation quantities, observables) than degrees of freedom. This is called super-integrability. The three-dimensional Hartmann potential has four motion integrals, so it has one more integral than degrees of freedom. Typical for super-integrable systems are:

Periodicity to quasi-periodicity of the classical trajectories or "random" quantum mechanical degeneration due to the energetic equivalence of the orbital angular momentum. Compared to the Kepler-Coulomb potential, this degeneracy is limited in the Hartmann potential.
Multi-separability, d. H. Separability of the classical Hamilton-Jacobi or the quantum mechanical Schrödinger equation in more than one orthogonal coordinate system. The Hartmann potential can be separated in terms of the spherical, the two spheroidal and the parabolic coordinates, so it is separable in four "rotating" coordinate systems.
Algebraic spectrum generation: For every exactly solvable potential there is a spectrum-generating algebra. The spectrum of the Hartmann potential is generated by an algebra of formal power series, the Hahn algebra .
q-Deformation: A deformed oscillator algebra can be assigned to every superintegratable system, i. H. the dynamic symmetry of the system can be made more flexible by one or more parameters, which is equivalent to making full use of the possibilities of symmetry. This fact is particularly important for understanding the behavior of matter because molecules are generally not rigid structures, but rather react flexibly to their surroundings. The algebraic structure of the symmetry flexibilization is called a quantum group or quantum algebra. Mathematically, this means examining the largest universal family of objects that are related to a reference object. The other elements of the family are then called the deformations of the reference object. The basic properties of the whole family are independent of the special nature of the reference object.

Because of the relationship that exists between superintegratability and purely number-theoretical structures, the understanding of exactly solvable potentials ultimately requires a self-interacting field.

Supersymmetry: the most general dynamic symmetry of the Hartmann potential

Because of their spherical symmetry, ring-shaped potentials like the Hartmann potential can be reduced to one-dimensional potentials. In one-dimensional systems, the most general dynamic symmetry is supersymmetry. The supersymmetry links the connecting, “bosonic” orientations of behavior (even operators) with the structuring, “fermionic” orientations (odd operators). Supersymmetry is therefore the instrument to penetrate into the unified superstructure of the behavior of matter; in particular, supersymmetry is the most general symmetry of the S matrix.

All characteristics of exactly solvable potentials can be understood as expressions of the underlying supersymmetric structure:

Super potential: In contrast to ordinary quantum mechanics, where the interrelationship of the degrees of freedom is recorded by potential energies, in the supersymmetric Hamilton operator the superpotential describes the internal connection of all possible orientations of behavior (spin and statistics). The super potential has the unit "square root of energy". The super potential of the bound states of the Hartmann potential links different angular impulses. In the radius range from zero to infinity the superpotential of the Hartmann system differs structurally from the superpotential of the Kepler-Coulomb potential only by the value of the constant.
Factorization: In one-dimensional systems, the supersymmetric Hamilton operator can always be factored by a pair of Hamilton operators, which are referred to as supersymmetric partners. So supersymmetry automatically includes the factorization method. The associated partner potentials are then the potentials with the dimension of energy (potential energy).
Sequential unfolding: Because the spectrum of partner Hamilton operators always differs by exactly one state, finite chains of bound states arise through sequential pair formation. Such supersymmetric chains describe a step-by-step one-dimensional unfolding process to increasingly complicated forms of excitation. The supersymmetrical partner formation also allows the expansion to multi-dimensional systems, which leads to new potentials with different eigenfunctions but the same spectrum as the initial potential. The dynamic development process is driven by the interrelationship between bosonic and fermionic alignment d. H. ultimately through the self-interaction of a superfield.
Shape invariance: Through the concept of form invariance ( Darboux transformation ), the class of potentials that can be solved exactly and in closed form can be precisely specified within the framework of supersymmetric quantum mechanics. For all exactly solvable potentials, the partner potentials are form-invariant with regard to the translation. Shape invariance is a sufficient condition for exact solvability. The Hartmann potential fulfills this condition. With the help of the supersymmetrical concept of form invariance, the spectrum of all exactly solvable one-dimensional potential types, there are only 10 in total, can be traced back to three basic forms: The harmonic oscillator is an example for the first form, the Kepler-Coulomb potential an example for the second and the box potential is an example of the third form. Due to the supersymmetric quantum mechanics, the exact solvability of potentials like the Hartmann potential is traced back to the shape invariance.

Due to its one-dimensionality as the primary area of validity, supersymmetry can be described as the “language level” of behavior if language is understood to be an interval-structured linear sequence that describes the development of increasingly complex forms.

The program of unification in the direction of a unified field of chemical bonding, which began with the Hartmann potential, is realized when the structure of the DNA and its “language”, the genetic code, is understood as the development of the self-interaction of a unified field.

Web links

P. Tempesta on exactly solvable potentials and number theory ( MS PowerPoint ; 1.0 MB)

literature

H. Hartmann: A new quantum mechanical treatment of CH ₄ and NH ₄⁺ . In: Journal of scientific research A . tape 2 , no. September 9 , 1947, p. 489-491 , doi : 10.1515 / zna-1947-0903 .

H. Hartmann, K. Jug: Application of a single-center method to the π-electron systems of five-heterocycles . In: Theoretica chimica acta . tape 3 , no. 5 , 1965, ISSN 0040-5744 , pp. 439-457 , doi : 10.1007 / BF00530422 .

H. Hartmann: New wave mechanical eigenvalue problems . Steiner-Verlag, Wiesbaden 1972, DNB 730209873 .

M. Kibler, T. Negadi: Motion of a particle in a ring-shaped potential: An approach via a nonbijective canonical transformation . In: International Journal of Quantum Chemistry . tape 26 , no. 3 , 1984, ISSN 1097-461X , pp. 405-410 , doi : 10.1002 / qua.560260308 .

GG Blado: Supersymmetry and the Hartmann Potential of Theoretical Chemistry . In: Theoretica Chimica Acta . tape 94 , 1996, pp. 53 , arxiv : quant-ph / 9602005 .

Chronological overview

This list of works on the Hartmann potential and its relationship to other potentials, its exact solvability and the underlying dynamic symmetry offers a chronological overview of the development in this area of theoretical chemistry:

H. Hartmann: The movement of a body in a ring-shaped potential field . In: Theoretica chimica acta . tape 24 , no. 2-3 , 1972, pp. 201-206 , doi : 10.1007 / BF00641399 .
H. Hartmann, R. Schuck, J. Radtke: The diamagnetic susceptibility of a non-spherically symmetric system . In: Theoretica chimica acta . tape 42 , no. 1 , 1976, p. 1-3 , doi : 10.1007 / BF00548285 .
H. Hartmann, D. Schuch: Spin-orbit coupling for the motion of a particle in a ring-shaped potential . In: International Journal of Quantum Chemistry . tape 18 , no. 1 , 1980, p. 125-141 , doi : 10.1002 / qua.560180119 .
M. Kibler, T. Negadi: Coulombic and ring-shaped potentials treated in a unified way via a nonbijective canonical transformation . In: Theoretica chimica acta . tape 66 , no. 1 , 1984, p. 31-42 , doi : 10.1007 / BF00577137 .
M. Kibler, P. Winternitz: Dynamical invariance algebra of the Hartmann potential . In: Journal of Physics A: Mathematical and General . tape 20 , no. 13 , 1987, pp. 4097 , doi : 10.1088 / 0305-4470 / 20/13/018 .
M. Kibler, P. Winternitz: Periodicity and Quasi-Periodicity for Super-Integrable Hamiltonian Systems . In: Physics Letters A . tape 147 , no. 7 , 1990, pp. 338-342 , arxiv : quant-ph / 0405017 .
V. Lutsenko, GS Pogosyan, AN Sisakyan, VM Ter-Antonyan: Hydrogen atom as indicator of hidden symmetry of a ring-shaped potential . In: Theoretical and Mathematical Physics . tape 83 , no. 3 , 1990, ISSN 0040-5779 , pp. 633-639 , doi : 10.1007 / BF01018033 .
YI Granovskii, AS Zhedanov, IM Lutzenko: Quadratic algebra as a 'hidden' symmetry of the Hartmann potential . In: Journal of Physics A: Mathematical and General . tape 24 , no. 16 , 1991, pp. 3887 , doi : 10.1088 / 0305-4470 / 24/16/024 .
M. Kibler, G ..- H. Lamot, P. Winternitz: Classical trajectories for two ring-shaped potentials . In: International Journal of Quantum Chemistry . tape 43 , no. 5 , 1992, pp. 625-645 , arxiv : quant-ph / 9810006 .
ED Filho: Supersymmetric quantum mechanics and two-dimensional systems . In: Brazilian Journal of Physics . tape 22 , no. 1 , 1992, p. 45-48 ( PDF ).
M. Kibler, C. Campigotto: Classical and quantum study of a generalized Kepler – Coulomb system . In: International Journal of Quantum Chemistry . tape 45 , no. 2 , 1993, p. 209-224 , doi : 10.1002 / qua.560450207 .
AS Zhedanov: Hidden symmetry algebra and overlap coefficients for two ring-shaped potentials . In: Journal of Physics A: Mathematical and General . tape 26 , no. 18 , 1993, p. 4633 , doi : 10.1088 / 0305-4470 / 26/18/027 .
M. Kibler, LG Mardoyan, GS Pogosyan: On a Generalized Kepler-Coulomb System: Interbasis Expansions . In: Int. J. Quantum Chem . tape 52 , 1994, pp. 1301-1316 , arxiv : hep-th / 9409033 .
D. Bonatsos, C. Daskaloyannis, K. Kokkotas: Deformed oscillator algebras for two dimensional quantum superintegrable systems . In: Physical Review A . tape 50 , no. 5 , 1994, pp. 3700-3709 , doi : 10.1103 / PhysRevA.50.3700 , arxiv : hep-th / 9309088 .
C. Grosche, GS Pogosyan, AN Sissakian : Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- and Three Dimensional Euclidean Space . In: Progress of Physics / Progress of Physics . tape 43 , no. 6 , 1995, pp. 453-521 , doi : 10.1002 / prop.2190430602 .
C. Grosche, GS Pogosyan, AN Sissakian : Path Integral Discussion for Smorodinsky-Winternitz Potentials: II. The Two- and Three-Dimensional Sphere . In: Progress of Physics / Progress of Physics . tape 43 , no. 6 , 1995, pp. 523-563 , doi : 10.1002 / prop.2190430603 .
D. Chang, W.-F. Chang: On Exactly Solvable Potentials . In: Chinese Journal of Physics . tape 33 , 1995, pp. 493-504 , arxiv : hep-th / 9509008 .
GG Blado: Supersymmetry and the Hartmann Potential of Theoretical Chemistry . In: Intern. J. Quantum Chemistry . tape 58 , 1996, pp. 431-439 , arxiv : quant-ph / 9602005 .
W. Miller: Multiseparability and superintegrability for classical and quantum systems. IMA Preprint Series, University of Minnesota, 1999
EG Kalnins, W. Miller, GS Pogosyan: Coulomb oscillator duality in spaces of constant curvature . In: Journal of Mathematical Physics . tape 41 , no. 5 , 2000, ISSN 0022-2488 , p. 2629-2657 , doi : 10.1063 / 1.533263 , arxiv : quant-ph / 9906055 .
A. Gangopadhyay, JV Mallow, C. Rasinariu, UP Sukhatme: Exact solutions of the Schroedinger Equation: Connection between supersymmetric quantum mechanics and spectrum generating algebras . In: Chinese Journal of Physics . tape 39 , 2001, p. 101-121 ( PDF ).
T. Bartsch: The Kustaanheimo-Stiefel transformation in geometric algebra . In: Journal of Physics A: Mathematical and General . tape 36 , no. 25 , 2003, p. 6963-6978 , doi : 10.1088 / 0305-4470 / 36/25/305 , arxiv : physics / 0301017 .

SM Ikhdair, R. Sever: Exact solutions of the radial Schrödinger equation for some physical potentials . In: Open Physics . tape 5 , no. 4 , 2007, ISSN 2391-5471 , p. 516-527 , doi : 10.2478 / s11534-007-0022-9 .

PR Giri: Self-Adjointness of Generalized MIC-Kepler System . In: Modern Physics Letters A . tape 22 , no. 31 , 2007, p. 2365-2377 , doi : 10.1142 / S0217732307022530 , arxiv : hep-th / 0603226v3 .

PR Giri: Supersymmetric quantum mechanical generalized MIC-Kepler system . In: Modern Physics Letters A . tape 23 , no. 12 , 2008, p. 895-904 , doi : 10.1142 / S0217732308025462 , arxiv : hep-th / 0607059 .

J. Sadeghi, B. Pourhassan: Exact Solution of the non-central modified Kratzer potential plus a ring-shaped like potential by the factorization method . In: Electronic Journal of Theoretical Physics . tape 5 , no. 17 , 2008, p. 197-206 ( PDF ).