Trygve Ulf Helgaker

• Dr. Philos.
• University of Oslo

The basis-set convergence of the electronic correlation energy in the water molecule is investigated at the second-order Mo&slash;ller–Plesset level and at the coupled-cluster singles-and-doubles level with and without perturbative triples corrections applied. The basis-set limits of the correlation energy are established to within 2 mEh by means o...

Elementary chemistry distinguishes two kinds of strong bonds between atoms in molecules: the covalent bond, where bonding arises from valence electron pairs shared between neighboring atoms, and the ionic bond, where transfer of electrons from one atom to another leads to Coulombic attraction between the resulting ions. We present a third, distinct...

Using a recently implemented technique for the calculation of the adiabatic connection (AC) of density functional theory (DFT) based on Lieb maximization with respect to the external potential, the AC is studied for atoms and molecules containing up to ten electrons: the helium isoelectronic series, the hydrogen molecule, the beryllium isoelectroni...

Traditionally, excitation energies in coupled-cluster (CC) theory have been calculated by solving the CC Jacobian eigenvalue equation. However, based on our recent work [Jørgensen et al., Sci. Adv. 10, eadn3454 (2024)], we propose a reformulation of the calculation of excitation energies where excitation energies are determined as a conventional mo...

Classical reaction barriers in density-functional theory are considered from the perspective of the density-fixed adiabatic connection. A ‘reaction adiabatic-connection integrand’, , is introduced, where λ is the electron–electron interaction strength, for which equals the barrier, meaning the barrier can be easily visualized as the area under a pl...

Non-adiabatic coupling matrix elements (NACMEs) are important in quantum chemistry, particularly for molecular dynamics methods such as surface hopping. However, NACMEs are gauge dependent. This presents a difficulty for their calculation in general, where there are no restrictions on the gauge function except that it be differentiable. These cases...

Non-adiabatic coupling matrix elements (NACMEs) are important in quantum chemistry, particularly for molecular dynamics methods such as surface hopping. However, NACMEs are gauge dependent. This presents a difficulty for their calculation in general, where there are no restrictions on the gauge function except that it be differentiable. These cases...

Conventional quantum-mechanical calculations of molecular properties, such as dipole moments and electronic excitation energies, give errors that depend linearly on the error in the wave function. An exception is the electronic energy, whose error depends quadratically on the error in wave function. We here describe how all properties may be calcul...

The exchange-only virial relation due to Levy and Perdew is revisited. Invoking the adiabatic connection, we introduce the exchange energy in terms of the right-derivative of the universal density functional w.r.t. the coupling strength λ at λ = 0. This agrees with the Levy–Perdew definition of the exchange energy as a high-density limit of the ful...

Ab initio molecular dynamics in a magnetic field requires solving equations of motion with velocity-dependent forces -- namely, the Lorentz force arising from the nuclear charges moving in a magnetic field and the Berry force arising from the shielding of these charges from the magnetic field by the surrounding electrons. In this work, we revisit t...

A semiclassical theory of small oscillations is developed for nuclei that are subject to velocity-dependent forces in addition to the usual interatomic forces. When the velocity-dependent forces are due to a strong magnetic field, novel effects arise—for example, the coupling of vibrational, rotational, and translational modes. The theory is first...

In an ultrastrong magnetic field, with field strength B ≈ B 0 = 2.35 × 10 ⁵ T, molecular structure and dynamics differ strongly from that observed on the Earth. Within the Born–Oppenheimer (BO) approximation, for example, frequent (near) crossings of electronic energy surfaces are induced by the field, suggesting that nonadiabatic phenomena and pro...

The Berry curvature is essential in Born-Oppenheimer molecular dynamics, describing the screening of the nuclei by the electrons in a magnetic field. Parts of the Berry curvature can be understood as the external magnetic field multiplied by an effective charge so that the resulting Berry force behaves like a Lorentz force during the simulations. H...

The Born-Oppenheimer (BO) approximation is less accurate in the presence of a strong magnetic field than in the absence of a field. This is due to the complicated and unpredictable response of electronic structure to the field, especially in the mixed regime $B \approx B_0 = 2.35 \times 10^5\,$T. Therefore, it is desirable to explore non-BO methods...

A semiclassical theory of small oscillations is developed for nuclei that are subject to velocity-dependent forces in addition to the usual interatomic forces. When the velocity-dependent forces are due to a strong magnetic field, novel effects arise -- for example, the coupling of vibrational, rotational, and translational modes. The theory is fir...

The Berry curvature is essential in Born$-$Oppenheimer molecular dynamics, describing the screening of the nuclei by the electrons in a magnetic field. Parts of the Berry curvature can be understood as the external magnetic field multiplied by an effective charge so that the resulting Berry force behaves like a Lorentz force during the simulations....

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchan...

The Berry connection and curvature are key components of electronic structure calculations for atoms and molecules in magnetic fields. They ensure the correct translational behavior of the effective nuclear Hamiltonian and the correct center-of-mass motion during molecular dynamics in these environments. In this work, we demonstrate how these prope...

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchan...

The Berry connection and curvature are key components of electronic structure calculations for atoms and molecules in magnetic fields. They ensure the correct translational behavior of the effective nuclear Hamiltonian and the correct center-of-mass motion during molecular dynamics in these environments. In this work, we demonstrate how these prope...

Molecular rotations and vibrations have been extensively studied by chemists for decades, both experimentally using spectroscopic methods and theoretically with the help of quantum chemistry. However, the theoretical investigation of molecular rotations and vibrations in strong magnetic fields requires computationally more demanding tools. As such,...

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchan...

Lieb's convex formulation of density-functional theory is presented in a pedagogical manner, emphasizing its connection to Hohenberg-Kohn theory and to Levy's constrained-search theory. The Hohenberg-Kohn and Lieb variation principles are discussed, highlighting the dual relationship between the ground-state energy and the universal density functio...

The diagonal nonadiabatic term arising from the Born-Oppenheimer wave function ansatz contains contributions from a vector and scalar potential. The former is provably zero when the wave function can be taken to be real valued, and the latter, known as the diagonal Born-Oppenheimer correction (DBOC), is typically small in magnitude. Therefore, unle...

The dynamics of a molecule in a magnetic field is significantly different from its zero-field counterpart. One important difference in the presence of a field is the Lorentz force acting on the nuclei, which can be decomposed as the sum of the bare nuclear Lorentz force and a screening force due to the electrons. This screening force is calculated...

Strong magnetic fields have a large impact on the dynamics of molecules. In addition to the changes in the electronic structure, the nuclei are exposed to the Lorentz force with the magnetic field being screened by the electrons. In this work, we explore these effects using ab initio molecular dynamics simulations based on an effective Hamiltonian...

Strong magnetic fields have a large impact on the dynamics of molecules. In addition to the changes of the electronic structure, the nuclei are exposed to the Lorentz force with the magnetic field being screened by the electrons. In this work, we explore these effects using ab-initio molecular dynamics simulations based on an effective Hamiltonian...

The dynamics of a molecule in a magnetic field is significantly different form its zero-field counterpart. One important difference in the presence of a field is the Lorentz force acting on the nuclei, which can be decomposed as the sum of the bare nuclear Lorentz force and a screening force due to the electrons. This screening force is calculated...

A cornerstone of current–density functional theory (CDFT) in its paramagnetic formulation is proven. After a brief outline of the mathematical structure of CDFT, the lower semicontinuity and expectation-valuedness of the CDFT constrained-search functional is proven, meaning that there is always a minimizing density matrix in the CDFT constrained-se...

We present a Gaussian-basis implementation of orbital-free density-functional theory (OF-DFT) in which the trust-region image method (TRIM) is used for optimization. This second-order optimization scheme has been constructed to provide benchmark all-electron results with very tight convergence of the particle-number constraint, associated chemical...

We present a Gaussian-basis implementation of orbital-free density-functional theory (OF-DFT) in which the trust-region image method (TRIM) is used for optimization. This second-order optimization scheme has been constructed to provide benchmark all-electron results with very tight convergence of the particle number constraint, associated chemical...

Interactions of a stationary external magnetic field with the spin and orbital magnetic momenta of a molecule are included in the quantum mechanical model where the Born–Oppenheimer approximation is not assumed. The model is used to calculate some of the lowest-lying internal bound states of the molecule for various strengths of the magnetic field....

We investigate the helium dimer in strong magnetic fields, focusing on the spectrum of low-lying electronic states and their dissociation curves, at the full configuration-interaction level of theory. To address the loss of cylindrical symmetry and angular momentum as a good quantum number for nontrivial angles between the bond axis and magnetic fi...

The Dalton Project provides a uniform platform access to the underlying full-fledged quantum chemistry codes Dalton and LSDalton as well as the PyFraME package for automatized fragmentation and parameterization of complex molecular environments. The platform is written in Python and defines a means for library communication and interaction. Interme...

We investigate the helium dimer in strong magnetic fields, focusing on the spectrum of low-lying electronic states and their dissociation curves, at the full configuration-interaction level of theory. To address the loss of cylindrical symmetry and angular momentum as a good quantum number for nontrivial angles between the bond axis and magnetic fi...

A suite of tools for the analysis of magnetically induced currents is introduced. These are applicable to both the weak-field regime, well described by linear response perturbation theory, and to the high-field regime, which is inaccessible to such methods. A disc-based quadrature scheme is proposed for the analysis of magnetically induced current...

DIRAC18 DIRAC, a relativistic ab initio electronic structure program, Release DIRAC18 (2018), written by T. Saue, L. Visscher, H. J. Aa. Jensen, and R. Bast, with contributions from V. Bakken, K. G. Dyall, S. Dubillard, U. Ekström, E. Eliav, T. Enevoldsen, E. Faßhauer, T. Fleig, O. Fossgaard, A. S. P. Gomes, E. D. Hedegård, T. Helgaker, J. Henrikss...

Quasiparticle energies of the atoms H-Ne have been computed in the GW approximation in the presence of strong magnetic fields with field strengths varying from 0 to 0.25 atomic units (0.25 B0=0.25 ℏe-1a0-2≈58 763 T). The GW quasiparticle energies are compared with equation-of-motion ionization-potential (EOM-IP) coupled-cluster singles-and-doubles...

Recent work has established Moreau-Yosida regularization as a mathematical tool to achieve rigorous functional differentiability in density-functional theory. In this article, we extend this tool to paramagnetic current-density-functional theory, the most common density-functional framework for magnetic field effects. The extension includes a well-...

Recent work has established Moreau-Yosida regularization as a mathematical tool to achieve rigorous functional differentiability in density-functional theory. In this article, we extend this tool to paramagnetic current-density-functional theory, the most common density-functional framework for magnetic field effects. The extension includes a well-...

A model for describing the states of a molecular system trapped in a cavity created by a fast-rotating strong magnetic field is proposed and implemented. All-particle explicitly correlated Gaussian functions with shifted centers are employed in the model to expand the wave functions of the system. Both “internal” states associated with the system’s...

A variety of approaches are presented for the computation of atomic and molecular correlation energies based on the Bethe-Salpeter equation in the framework of the adiabatic-connection fluctuation-dissipation theorem. The performance of the approaches is assessed by computing the total energies of the atoms H-Ne and the atomization energies of the...

We study the total molecular electronic energy and its Kohn–Sham components within the framework of magnetic-field density-functional theory (BDFT), an alternative to current-dependent density-functional theory (CDFT) for molecules in the presence of magnetic fields. For a selection of closed-shell dia- and paramagnetic molecules, we investigate th...

A detailed account of the Kohn-Sham algorithm from quantum chemistry, formulated rigorously in the very general setting of convex analysis on Banach spaces, is given here. Starting from a Levy-Lieb-type functional, its convex and lower semi-continuous extension is regularized to obtain differentiability. This extra layer allows to rigorously introd...

A detailed account of the Kohn-Sham algorithm from quantum chemistry, formulated rigorously in the very general setting of convex analysis on Banach spaces, is given here. Starting from a Levy-Lieb-type functional, its convex and lower semi-continuous extension is regularized to obtain differentiability. This extra layer allows to rigorously introd...

A Görling–Levy (GL)-based perturbation theory along the range-separated adiabatic connection is assessed for the calculation of electronic excitation energies. In comparison with the Rayleigh–Schrödinger (RS)-based perturbation theory this GL-based perturbation theory keeps the ground-state density constant at each order and thus gives the correct...

A Görling-Levy (GL)-based perturbation theory along the range-separated adia-batic connection is assessed for the calculation of electronic excitation energies. In comparison with the Rayleigh-Schrödinger (RS)-based perturbation theory introduced in a previous work [E. Rebolini, J. Toulouse, A. M. Teale, T. Helgaker, A. Savin, Mol. Phys. 113, 1740...

A G{\"o}rling-Levy (GL)-based perturbation theory along the range-separated adiabatic connection is assessed for the calculation of electronic excitation energies. In comparison with the Rayleigh-Schr{\"o}dinger (RS)-based perturbation theory introduced in a previous work [E. Rebolini, J. Toulouse, A. M. Teale, T. Helgaker, A. Savin, Mol. Phys. 113...

A recently proposed variation principle [N. I. Gidopoulos, Phys. Rev. A 83, 040502(R) (2011)] for the determination of Kohn–Sham effective potentials is examined and extended to arbitrary electron-interaction strengths and to mixed states. Comparisons are drawn with Lieb’s convex-conjugate functional, which allows for the determination of a potenti...

We construct a density-functional formalism adapted to uniform external magnetic fields that is intermediate between conventional Density Functional Theory and Current-Density Functional Theory (CDFT). In the intermediate theory, which we term LDFT, the basic variables are the density, the canonical momentum, and the paramagnetic contribution to th...

We construct a density-functional formalism adapted to uniform external magnetic fields that is intermediate between conventional Density Functional Theory and Current-Density Functional Theory (CDFT). In the intermediate theory, which we term LDFT, the basic variables are the density, the canonical momentum, and the paramagnetic contribution to th...

We study the effects of magnetic fields in the context of magnetic field density-functional theory (BDFT), where the energy is a functional of the electron density ρ and the magnetic field B. We show that this approach is a worthwhile alternative to current-density functional theory (CDFT) and may provide a viable route to the study of many magneti...

Explicitly correlated all-particle Gaussian functions with shifted centers (ECGs) are implemented within the earlier proposed effective variational non-Born–Oppenheimer method for calculating bound states of molecular systems in magnetic field (CPL 639, 295 (2015)). The Hamiltonian used in the calculations is obtained by subtracting the operator re...

An atomic orbital density matrix based response formulation of the nuclei-selected approach of Beer, Kussmann, and Ochsenfeld [J. Chem. Phys. 134, 074102 (2011)] to calculate nuclear magnetic resonance (NMR) shielding tensors has been developed and implemented into LSDalton allowing for a simultaneous solution of the response equations, which signi...

We compare the NMR indirect nuclear spin-spin coupling constants in strychnine calculated using density-functional theory (DFT) with the semi-empirical relativistic-force-field (RFF) method of Kutateladze and Mukhina (KM, J. Org. Chem., 2015, 80, 10838-10848). Significantly more accurate DFT values than those obtained by KM for their comparison wit...

We compare the performance of three approximate methods for speeding up evaluation of the exchange contribution in Hartree–Fock and hybrid Kohn–Sham calculations: the chain-of- spheres algorithm [COSX, Neese et al. Chem. Phys. 2008, 356, 98-109], the pair-atomic resolution-of-identity method [PARI-K, Merlot et al. J. Comput. Chem. 2013, 34, 1486-14...

We present a generalisation of the electron localisation function (ELF) to current-density-functional theory as a descriptor for the properties of molecules in the presence of magnetic fields. The resulting current ELF (cELF) is examined for a range of small molecular systems in field strengths up to B0 = 235 kT (one atomic unit). The cELF clearly...

DIRAC, a relativistic ab initio electronic structure program, Release DIRAC16 (2016), written by H. J. Aa. Jensen, R. Bast, T. Saue, and L. Visscher, with contributions from V. Bakken, K. G. Dyall, S. Dubillard, U. Ekstroem, E. Eliav, T. Enevoldsen, E. Fasshauer, T. Fleig, O. Fossgaard, A. S. P. Gomes, T. Helgaker, J. Henriksson, M. Ilias, Ch. R. J...

We study the evaluation of the Gross-Oliveira-Kohn expression for excitation energies E 1 − E 0 = ε 1 − ε 0 + ∂ E x c , w [ ρ ] ∂ w | ρ = ρ 0 .

An effective variational non-Born-Oppenheimer method is applied to calculate the ground state of the HD molecule in a strong magnetic field. The Hamiltonian used in the calculations is obtained by subtracting the operator representing the kinetic energy of the center-of-mass motion from the total laboratory-frame Hamiltonian. Orbital basis sets are...

Plots of electronic energy vs electron number, determined using approximate density functional theory (DFT) and Hartree-Fock theory, are typically piecewise convex and piecewise concave, respectively. The curves also commonly exhibit a minimum and maximum, respectively, in the neutral → anion segment, which lead to positive DFT anion HOMO energies...

The adiabatic-connection framework has been widely used to explore the properties of the correlation energy in density-functional theory. The integrand in this formula may be expressed in terms of the electron–electron interactions directly, involving intrinsically two-particle expectation values. Alternatively, it may be expressed in terms of the...

The Tamm-Dancoff approximation (TDA) can be applied to the computation of excitation energies using time-dependent Hartree-Fock (TD-HF) and time-dependent density-functional theory (TD-DFT). In addition to simplifying the resulting response equations, the TDA has been shown to significantly improve the calculation of triplet excitation energies in...

The relationship between the densities of ground-state wave functions (i.e., the minimizers of the Rayleigh–Ritz variation principle) and the ground-state densities in density-functional theory (i.e., the minimizers of the Hohenberg–Kohn variation principle) is studied within the framework of convex conjugation, in a generic setting covering molecu...

The sources of error in the calculation of nuclear-magnetic-resonance shielding constants determined by density-functional theory are examined. Highly accurate Kohn–Sham wave functions are obtained from coupled-cluster electron density functions and used to define accurate—but current independent—density-functional shielding constants. These new re...

An implementation of coupled-cluster (CC) theory to treat atoms and molecules in finite magnetic fields is presented. The main challenges stem from the magnetic-field dependence in the Hamiltonian, or, more precisely, the appearance of the angular momentum operator, due to which the wave function becomes complex and which introduces a gauge-origin...

We present the self-consistent implementation of current-dependent (hybrid) meta generalized gradient approximation (mGGA) density functionals using London atomic orbitals. A previously proposed generalized kinetic energy density is utilized to implement mGGAs in the framework of Kohn--Sham current density-functional theory (KS-CDFT). A unique feat...

In this paper, an alternative method to range-separated linear-response time-dependent density-functional theory and perturbation theory is proposed to improve the estimation of the energies of a physical system from the energies of a partially interacting system. Starting from the analysis of the Taylor expansion of the energies of the partially i...

Abundance mass spectra, obtained upon carefully electrospraying solutions of tert-butanol (TB) in water into a mass spectrometer, display a systematic series of peaks due to mixed H(+)(TB)m(H2O)n clusters. Clusters with m + n = 21 have higher abundance (magic number peaks) than their neighbours when m ≤ 9, while for m > 9 they have lower abundance....

DIRAC, a relativistic ab initio electronic structure program, Release DIRAC15 (2015), written by R. Bast, T. Saue, L. Visscher, and H. J. Aa. Jensen, with contributions from V. Bakken, K. G. Dyall, S. Dubillard, U. Ekstroem, E. Eliav, T. Enevoldsen, E. Fasshauer, T. Fleig, O. Fossgaard, A. S. P. Gomes, T. Helgaker, J. Henriksson, M. Ilias, Ch. R. J...

We explore the possibility of calculating electronic excited states by using perturbation theory along a range-separated adiabatic connection. Starting from the energies of a partially interacting Hamiltonian, a first-order correction is defined with two variants of perturbation theory: a straight-forward perturbation theory, and an extension of th...

A new variation of the second order Generalized van Vleck Perturbation Theory (GVVPT2) for molecular electronic structure is suggested. In contrast to established procedure, in which CASSCF or MCSCF orbitals are first obtained and subsequently used to define a many-electron model (or reference) space, the use of an orbital space obtained from the L...

Three new variants of the auxiliary-density-matrix method (ADMM) of Guidon, Hutter, and VandeVondele [J. Chem. Theory Comput. 6, 2348 (2010)] are presented with the common feature that they have a simplified constraint compared with the full orthonormality requirement of the earlier ADMM1 method. All ADMM variants are tested for accuracy and perfor...

Second quantization formalism is introduced for an efficient description of molecular electronic systems in the nonrelativistic limit and an explicit description of electron spin. Spin orbitals are functions of three continuous spatial coordinates and one discrete spin coordinate. Quantum-mechanical operators may be classified according to how they...

Dynamical effects on the mechanochemistry of linear alkane chains, mimicking polyethylene, are studied by means of molecular dynamics simulations. Butane and oc- tane are studied using density-functional theory (DFT), whereas higher homologues are studied using a simple one-dimensional model in which the molecules are represented by a linear chain...

This chapter explores the mathematical properties and usefulness of the atomic basis functions by carrying out simple expansions of the ground-state orbitals of the carbon atom. Molecular orbitals may be constructed in one of two ways such as numerically or algebraically by linear expansion in a set of basis functions. The numerical approach offers...

This chapter discusses the relationship between exact wave functions and approximate wave functions, with emphasis on size-extensivity, the variation principle and symmetry restrictions. It describes important characteristic properties of the exact solution to the time-independent Schr?dinger equation for a molecular electronic system. The exact wa...

This chapter considers techniques for the evaluation of the one- and two-electron molecular integrals needed for the calculation of the standard ab initio wave functions and energies. Only integrals over GTOs will be considered, since these are the only basis functions widely used for multicentre molecular calculations. Having introduced the GTOs t...

This chapter begins with a general discussion of configuration-interaction(CI) expansions and configuration selection. It considers the problems associated with designing size-extensive configuration spaces. The chapter presents a simple model calculation to illustrate the size-extensivity problem of truncated CI wave functions. It then focuses on...

This chapter discusses a coupled-cluster method and a detailed exposition of the coupled-cluster singles-and-doubles (CCSD) model. It shows an important exponential ansatz of coupled-cluster theory and its optimization. The chapter also discusses various aspects of coupled-cluster theory such as size-extensivity and optimization techniques. It pres...

This chapter reviews functional form of Gaussian basis functions. It discusses Gaussian basis sets for Hartree-Fock calculations and correlated calculations. The chapter examines the convergence of the basis sets towards the basis-set limit. It considers the problems associated with basis-set superposition errors, in particular in calculations of i...

This chapter considers the unitary transformations of creation and annihilation operators and of Fock-space states that are generated by such transformations of the underlying spin-orbital basis. It shows how, in second quantization, the unitary transformations can be conveniently carried out by the exponential of an anti-Hermitian operator. The ch...

This chapter examines the more important standard models of quantum chemistry, comparing and characterizing the various models in relationship to one another and to the exact solution in Fock space. It considers the relationship between the expansions made in the one-particle space (the orbital space) and in the iV-particle space (the Fock space)....

This chapter presents the fundamentals of Rayleigh-Schrodinger perturbation theory (RSPT), deriving formulae for the energy and the wave function to arbitrary orders and discussing Wigner's 2n + 1 rule and Hylleraas functional and size-extensivity. It shows an exposition of M0ller-Plesset perturbation theory (MPPT) and the most successful applicati...

This chapter discusses the structure and construction of restricted Hartree-Fock wave functions. It also discusses second-order methods of optimization, based on an expansion of the Hartree-Fock energy in nonredundant orbital rotations, as well as density-based methods, required for the efficient application of Hartree-Fock theory to large molecula...

This chapter makes an extensive assessment of the methods, presenting investigations and comparisons of calculations made using standard basis sets and wave functions. Clearly, the usefulness of such investigations increases if they are performed for a large variety of molecular systems, comprising a number of different prototypical systems, and if...