Hans Hellmann

Hans Hellmann (1903–1938) was a German theoretical chemist who helped found quantum chemistry, originated the pseudopotential method for treating heavy atoms, and published the force theorem now known as the Hellmann–Feynman theorem. He worked at the Karpov Institute in Moscow from 1934 to 1937, was imprisoned by the Soviet secret service in 1938, and was executed a few weeks later.¹ His published achievements include clarifying the nature of the covalent chemical bond (1933), the molecular virial theorem (1933), the quantum-mechanical force theorem (1933, 1936/1937), the pseudopotential method (1934), and a theory of diabatic and adiabatic elementary reactions (1935) later reworked by Born and Huang.² Leibniz University Hannover records that he crossed the border between physics and chemistry and developed theorems for chemical bonding and reactions that remain valid and important today.³

Life and career: Germany, Moscow, Hannover

When the Nazis seized power in Germany in 1933, Hellmann realized that emigration became a necessity for his family.¹ In early 1934 he urgently needed a new position to support them. After a 1932 conversation in Berlin with Frumkin, deputy director of the Karpov Institute of Physical Chemistry in Moscow, he received a successful invitation from Academician Bach, the institute's director.⁴ He worked at the Karpov Institute from 1934 to 1937.¹

In Moscow he published under the Russified name Г. Гельман (G. Gel'man) in Russian-language journals such as Uspekhi Khimii in 1935–1936, while also placing English-language papers in the Journal of Chemical Physics: "A New Approximation Method in the Problem of Many Electrons", submitted from Moscow on 26 November 1934 and published in 1935, and a 1936 paper with W. Kassatotschkin on metallic binding according to the combined approximation procedure.⁶ Leibniz University Hannover records his work.³

Scientific contributions

The Heitler–London treatment of the covalent bond in H₂ (1927) is regarded as the birth of quantum chemistry, and Hellmann was among the pioneers who built molecular orbital and valence bond concepts in the following decade.⁴ His six "scientific masterpieces", as one historical account counts them, include the Hellmann–Feynman theorem, used in molecular geometry gradient optimization; pseudopotentials, which made possible the entry of quantum chemical treatment into the area of heavier atoms; work on non-covalent bonds; and the virial theorem.¹

Pseudopotentials. In 1934 Hellmann developed the idea of treating only the valence electrons quantum-mechanically, in the effective fields of the atomic cores, with the Pauli exclusion effect simulated by a repulsive potential.⁴ He developed the foundations of this method for molecules and solids under the name "combined approximation method", now one of the most important methods in quantum chemistry for calculating compounds of heavy atoms.⁷ His original form mimicked the repulsive, fermionic character of the inner electrons with an experimentally fitted, exponentially damped potential term, but it fails even for lithium; a 1974 study extended the idea with a second "shielded potential" term.⁸

Textbooks. In 1937 Hellmann wrote the first textbooks of the still very young field, one in Russian and one in German. The Russian Квантовая Химия was published by ONTI in Moscow and Leningrad, 546 pages, translated from the German manuscript by J. Golovin, N. Tunitskij, and M. Kovner; the German Einführung in die Quantenchemie appeared from Franz Deuticke Verlag in Leipzig and Vienna in March 1937, VIII + 350 pages, and was reviewed in 1938 by J. Syrkin and O. Schmidt.⁶ The Russian work is still considered an impressive textbook.¹ Unlike Pauling and Wilson (1935) or Eyring, Walter, and Kimball (1944), Hellmann's textbooks were not reprinted or reissued before the German edition was reprinted in 2015, a fact the publisher of the 2015 reprint calls insufficiently explained by his tragic fate.²

The Hellmann–Feynman theorem

The theorem states that a non-degenerate eigenvalue of a hermitian operator varies with a parameter according to

$$\frac{\partial E}{\partial \nu} = \left\langle \psi \left| \frac{\partial H}{\partial \nu} \right| \psi \right\rangle,$$

provided the normalized eigenfunction is continuous in the parameter.¹⁴ In 1937 Hellmann published the force expression for two atoms,

$$K = -\frac{dE(R)}{dR} = \frac{Z_1 Z_2}{R^2} - \int \frac{d r_i \cdot \rho(r_i) \cdot Z_i z_i}{r_i^3},$$

which Feynman also found in 1939. At equilibrium $K = 0$; during a reaction or a vibration, it is generally nonzero.⁴ The electronic contribution to the force depends on the electronic density, while the total force also includes the nuclear-nuclear repulsion term; the nuclear-coordinate-dependent Hamiltonian terms are the nuclear-nuclear repulsion and the electron-nuclear attraction.⁹ Properties summarized under the theorem include the virial and the electrostatic theorems.¹⁰

In practice, with atom-centered basis sets the naive Hellmann–Feynman force must be corrected for the Pulay term, which arises because the basis moves with the nuclei; if naively discarded it typically produces force errors upwards of 10 eV/Å. A 2022 study showed that augmented Gaussian basis sets suppress the Pulay force so that Hellmann–Feynman forces match analytic forces in density functional calculations.⁹

Attribution and priority: Hellmann, Feynman, Güttinger

Feynman discovered the theorem while he was an undergraduate student.¹⁰ Neither Feynman nor Hellmann was first to prove it. Paul Güttinger, a protégé of Wolfgang Pauli, may have been the first to publish a careful derivation, and precursors had appeared at least as early as 1922, with versions attributed to Pauli (1922), Schrödinger (1926), and Born and Fock.¹⁴ The canonical references remain Hellmann's 1937 Einführung in die Quantenchemie and Feynman's 1939 Physical Review 56, 340 paper.¹¹ The naming asymmetry reflects fame rather than priority: Richard P. Feynman is widely known, but Hellmann is relatively unknown.¹⁴

Arrest, execution, and rehabilitation

Hellmann was imprisoned in 1938 and executed a few weeks later by the Soviet secret service.¹ The investigation file on him, held in the archive of the Federal Security Service (FSB) in Moscow and examined and translated by Sabine Arnold, contains a personal data sheet, interrogation protocols, Hellmann's own signed statement, the indictment, a document on his execution, archiving confirmation, and rehabilitation documents.⁴ The file notes that there was no evidence beyond Hellmann's own statement and the interrogation protocols; in the rehabilitation proceedings, Academician Betrjanov and former colleague Ljubarskij described Hellmann as loyal to the Soviet Union and an anti-fascist.⁴ Colleagues of those arrested by the secret police publicly distanced themselves from the arrested, a pattern that shaped how his case was handled.⁴

Legacy and modern use

The pseudopotential method became popular only a generation later, when it was propagated in simplified form in the Anglo-Saxon world; most quantum-chemical calculations on molecules, clusters, and crystals containing medium-heavy or heavy atoms are now carried out with such pseudopotentials.⁴ The relativistic pseudopotential approximation is now the most widely applied method for systems containing heavy elements, and modern relativistic pseudopotentials adjusted to valence spectra give results in excellent agreement with all-electron methods when a small-core definition is used.⁵

Hellmann–Feynman forces remain central to force and geometry work. A 2022 machine-learning pipeline predicts the electronic density and obtains forces via the Hellmann–Feynman theorem, reducing training-data cost because computing the density from first principles is much simpler than computing the analytic force.⁹ Integrated and integral Hellmann–Feynman formulas, developed in 1967, relate expectation-value differences for exact molecular wavefunctions and remain applicable with approximate wavefunctions.¹²

Commemoration and recovery followed decades after his death. Philipps-Universität Marburg holds a Hans-Hellmann-Lecture named for him.⁷ Springer Spektrum reprinted Einführung in die Quantenchemie in 2015 with biographical notes by his son Hans Hellmann Jr.^{2 • 7} The 1999 multi-part biographical study by W.H.E. Schwarz, D. Andrae, S.R. Arnold, J. Heidberg, and colleagues, with revisions by J. Hinze and A. Karachalios, reconstructed his life and work.⁶

Three of the most important pioneers of modern theoretical chemistry, Hund, Mulliken, and Hückel, were all born in 1896; Hückel never received due recognition while active.¹³

References

1. Hans Hellmann: Life Story of a Scientist in the 20th Century, Chemické listy
2. Hans Hellmann: Einführung in die Quantenchemie, Springer reprint with biographical notes
3. Hans Hellmann, Research@Leibniz University Hannover
4. Pionier der modernen Quantenchemie: Hans Hellmann, Chemie in unserer Zeit (2004)
5. The Pseudopotential Approximation in Electronic Structure Theory, ChemPhysChem (2011)
6. List of publications of Hans Hellmann, D. Andrae, FU Berlin
7. Hans-Hellmann-Lecture, Philipps-Universität Marburg
8. An improved Hellmann-type pseudopotential for atoms and molecules, Int. J. Quantum Chem. (1974)
9. Accurate Hellmann–Feynman forces from density functional calculations with augmented Gaussian basis sets, arXiv (2022)
10. Hellmann-Feynman Theorem, ScienceDirect Topics
11. Hellmann–Feynman forces & the Pulay term, The Grid methods reference
12. Integrated and Integral Hellmann—Feynman Formulas, J. Chem. Phys. 47, 1275 (1967)
13. What I like about Hückel theory, J. Comput. Chem. (2006)
14. An Introduction to Hellmann-Feynman Theory, University of Central Florida
15. Hans G. Hellmann (1903 - 1938): ein Pionier der Quantenchemie, publish.UP (University of Potsdam)

Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Chemists › Researchers in physical, theoretical, and computational chemistry › Quantum chemistry and electronic structure theory

Initially written Oct 10, 2026 · Edited: Oct 15, 2026; Oct 21, 2026; Oct 25, 2026

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