London Orbitals Research Articles

Magnetic-translational sum rule and approximate models of the molecular Berry curvature.

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 properties of the Berry connection and curvature arise from the translational symmetry of the electronic wave function and how they are fully captured by a finite basis set of London orbitals but not by standard Gaussian basis sets. This is illustrated by a series of Hartree-Fock calculations on small molecules in different basis sets. Based on the resulting physical interpretation of the Berry curvature as the shielding of the nuclei by the electrons, we introduce and test a series of approximations using the Mulliken fragmentation scheme of the electron density. These approximations will be particularly useful in ab initio molecular dynamics calculations in a magnetic field since they reduce the computational cost, while recovering the correct physics and up to 95% of the exact Berry curvature.

The GW/BSE Method in Magnetic Fields.

The GW approximation and the Bethe–Salpeter equation have been implemented into the Turbomole program package for computations of molecular systems in a strong, finite magnetic field. Complex-valued London orbitals are used as basis functions to ensure gauge-invariant computational results. The implementation has been benchmarked against triplet excitation energies of 36 small to medium-sized molecules against reference values obtained at the approximate coupled-cluster level (CC2 approximation). Finally, a spectacular change of colour from orange to green of the tetracene molecule is induced by applying magnetic fields between 0 and 9,000 T perpendicular to the molecular plane.

Ab initio molecular dynamics with screened Lorentz forces. I. Calculation and atomic charge interpretation of Berry curvature.

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 from the Berry curvature and can change the dynamics qualitatively. It is therefore important to include the contributions from the Berry curvature in molecular dynamics simulations in a magnetic field. In this work, we present a scheme for calculating the Berry curvature numerically using a finite-difference technique, addressing challenges related to the arbitrary global phase of the wave function. The Berry curvature is calculated as a function of bond distance for H2 at the restricted and unrestricted Hartree-Fock levels of theory and for CH+ as a function of the magnetic field strength at the restricted Hartree-Fock level of theory. The calculations are carried out using basis sets of contracted Gaussian functions equipped with London phase factors (London orbitals) to ensure gauge-origin invariance. In this paper, we also interpret the Berry curvature in terms of atomic charges and discuss its convergence in basis sets with and without London phase factors. The calculation of the Berry curvature allows for its inclusion in ab initio molecular dynamics simulations in a magnetic field.

Relativistic Effects in Magnetic Circular Dichroism: Restricted Magnetic Balance and Temperature Dependence

Magnetic circular dichroism of transition metal complexes and open-shell systems are challenging to simulate and analyze, mainly due to the interplay of spin-orbit couplings and finite-magnetic-field included Zeeman effects with the complex selection rules dictated by the circularly polarized light. In this work, we introduce an ab initio relativistic two-component formalism based on the restricted magnetic-balanced Hamiltonian for simulating MCD spectra. Both homogeneous finite magnetic field and relativistic effects are included variationally in the ground state reference. Finite-field London orbitals are used to enforce the constrained gauge-origin independence in the calculation using localized atomic orbitals. Through benchmark studies of AuCl4-, Pt(CN)42-, and Mo(CN)83-, we discuss how relativistic effects are manifested in MCD for both closed-shell and open-shell molecular complexes and how the interplay between spin-orbit coupling and magnetic field modulates the MCD selection rules. Finally, an investigation on temperature-dependent MCD is carried out and compared to experiment.

Linking Optical and Magnetic Properties of (anti)Aromatic Porphyrinoids

Electronic and spectroscopic properties of porphyrinoid macrocycles are, to a large extent, determined by the number of π-electrons participating in the conjugation network. Full control of the number of π-electrons implies full control of the aromatic or antiaromatic character of the molecule. To achieve this, knowledge about the electron mobility pathways within a molecule is essential. Experimentally, these pathways are not easily accessible. Therefore theoretical studies are necessary to complement experiments that aim at systematically synthesizing porphyrinoids with distinct properties. A possible way to achieve this is the systematic investigation of magnetically induced current densities to determine reliable aromatic pathways in molecules.[1,2]In my talk I will present an overview of different ways to systematically calculate and analyze the magnetically induced current density in porphyrin based molecules. The computational approach used is the gauge including magnetically induced current density method (GIMIC).[3,4] GIMIC is a independent program that is used for the calculation of magnetically induced current densities employing London orbitals.[3,4] Numerical integration of the current flow around molecular rings and along selected chemical bonds can be used for determining current pathways and the degree of aromaticity of various molecules according to the magnetic criterion.[1-4] It is possible to calculate the anisotropy of the magnetically induced density using gauge including atomic orbitals with GIMIC.[5] It has also been found that the integrated current strength susceptibility and optical properties of antiaromatic porphyrinoids are related.[6,7] A new feature of GIMIC is the 3D visualization of the current density as streamlines including 3D animations of the current flow.[8-11] In this context very recent results will be highlited illustrating the potential and the challenges of the method.

Efficient evaluation of three-centre two-electron integrals over London orbitals

The nonperturbative calculation of molecular properties in magnetic fields requires the evaluation of integrals over complex-valued Gaussian-type London atomic orbitals (LAOs). With these orbitals, the calculation of four-centre electron-repulsion integrals (ERIs) is particularly demanding, because their permutational symmetry is lowered, and because complex algebra is required. We have implemented the resolution-of-the-identity (RI) approximation for LAOs in the TURBOMOLE program package. With respect to LAOs, employing the RI approximation is particularly beneficial, because the auxiliary basis set may always be chosen to be real-valued. As a consequence, the two-centre integrals in the RI approximation remain real-valued, and the three-centre integrals possess the same permutational symmetry as their real-valued counterparts. Compared to a direct calculation of four-centre ERIs over LAOs, using the RI approximation thus not only reduces the scaling of the integral evaluation, but also increases the efficiency by an additional factor of at least two. By using other well-established methods such as Cauchy–Schwarz screening, the difference-density approach, and Pulay's direct inversion in the iterative subspace (DIIS), the efficiency of nonperturbative calculations in magnetic fields can be increased even further.

Simulating Magnetic Circular Dichroism Spectra with Real-Time Time-Dependent Density Functional Theory in Gauge Including Atomic Orbitals.

Magnetic circular dichroism (MCD) spectra are able to provide insights into the geometric, electronic, and magnetic properties of chemical systems. However, they can be challenging to understand and simulate given the need to simultaneously treat both the finite magnetic and optical fields. Thus, efficient simulations are desired to understand the spectra and resolve the molecular electronic states. Real-time dynamics are used widely in the simulation of electronic spectroscopies such as absorption as well as electronic circular dichroism, but simulating MCD with real-time dynamics is technically and theoretically challenging. In this work, we introduce a real-time dynamics-based ab initio method with a nonperturbative treatment of a static magnetic field with London orbitals for simulating the MCD spectra of closed shell systems. Effects of a magnetic field are included variationally in the spin-free nonrelativistic Hamiltonian. Real-time time-dependent density functional theory dynamics are then performed, from which we compute the response function in the presence of the external magnetic field, giving the MCD spectrum. The method developed in this paper is applied to simulate the MCD spectra for pyrimidine, pyrazine, and 1,4-naphthoquinone. Results are discussed and compared to the experiment.

An ab Initio Linear Response Method for Computing Magnetic Circular Dichroism Spectra with Nonperturbative Treatment of Magnetic Field.

Magnetic circular dichroism (MCD) experiments provide a sensitive tool for exploring geometric, magnetic, and electronic properties of chemical complexes and condensed matter systems. They are also challenging to simulate because of the need to simultaneously treat the perturbations of a finite magnetic field as well as an optical field. In this work, we introduce an ab initio approach that treats the external magnetic field nonperturbatively with London orbitals for simulating the MCD spectra of closed-shell systems. Effects of a magnetic field are included variationally in the spin-free nonrelativistic Hamiltonian, followed by a linear response formalism to directly calculate the difference in absorption between the left and right circularly polarized light. In addition to the presentation of underlying mathematical formalism and implementation, the method developed in this paper has been applied to simulations of MCD spectra of the sodium anion, 2,2,6,6-tetramethylcyclohexanone, and 3-methyl-2-hexanone. Results are discussed and compared to experiments.

Generalized Hartree-Fock with Nonperturbative Treatment of Strong Magnetic Fields: Application to Molecular Spin Phase Transitions.

In this work, we present a framework of an ab initio variational approach to effectively explore electronic spin phase transitions in molecular systems inside of a homogeneous magnetic field. In order to capture this phenomenon, the complex generalized Hartree-Fock ([Formula: see text]) method is used in the spinor formalism with London orbitals. Recursive algorithms for computing the one- and two-electron integrals of London orbitals are also provided. A Pauli matrix representation of the [Formula: see text] method is introduced to separate spin contributions from the scalar part of the Fock matrix. Next, spin phase transitions in two different molecular systems are investigated in the presence of a strong magnetic field. Noncollinear spin configurations are observed during the spin phase transitions in H2 and a dichromium complex, [(H3N)4Cr(OH)2Cr(NH3)4]4+, with an increase in magnetic field strength. The competing driving forces of exchange coupling and the spin Zeeman effect have been shown to govern the spin phase transition and its transition rate. Additionally, the energetic contributions of the spin Zeeman, orbital Zeeman, and diamagnetic terms to the potential energy surface are also analyzed.

New Insights on Aromatic Pathways in Porphyrinoids

Electronic and spectroscopic properties of porphyrinoid macrocycles are, to a large extent, determined by the number of π-electrons participating in the conjugation network. Full control of the number of π-electrons implies full control of the aromatic or antiaromatic character of the molecule. To achieve this, knowledge about the electron mobility pathways within a molecule is essential. Experimentally, these pathways are not easily accessible. Therefore theoretical studies are necessary to complement experiments that aim at systematically synthesizing porphyrinoids with distinct properties. A possible way to achieve this is the systematic investigation of magnetically induced current densities to determine reliable aromatic pathways in molecules.[1] In my talk I will present an overview of different ways to systematically calculate and analyze the magnetically induced current density in porphyrin based molecules. The computational approach used in my group is the gauge including magnetically induced current density method (GIMIC).[2,3] GIMIC is a independent program that is used for the calculation of magnetically induced current densities using London orbitals.[2, 3] Numerical integration of the current flow around molecular rings and along selected chemical bonds can be used for determining current pathways and the degree of aromaticity of various molecules according to the magnetic criterion.[4,5] A new feature of the GIMIC program is the calculation of the anisotropy of the magnetically induced density using gauge including atomic orbitals.[6] Very recent results obtained for various porphyrinoid molecules will be highlighted illustrating the potential and challenges of the method.[7-9]

Gauge-origin independent calculations of electric-field-induced second-harmonic generation circular intensity difference using London atomic orbitals

ABSTRACTWe present the first gauge-origin independent calculations of the circular intensity difference (CID) in electric-field-induced second-harmonic generation (EFISHG), including all contributions up to the electric-quadrupole–magnetic-dipole level. A recursive, open-ended response theory framework in combination with the use of London atomic orbitals allows us to ensure gauge-origin independent results. We apply this approach to study EFISHG-CID in a collection of chiral amino acids. We demonstrate that diffuse polarising basis functions are critical in order to obtain accurate CIDs, and that a basis set of at least aug-cc-pVTZ quality is needed in order to obtain results close to the basis-set limit. The use of London orbitals does not lead to significantly faster basis set convergence, although the improved basis set convergence allows the aug-cc-pVDZ basis set to be used with some confidence for larger molecules.

(Invited) Predicting the Degree of Aromaticity of Novel Porphyrinoid Compounds

Electronic and spectroscopic properties of porphyrinoid macrocycles are, to a large extent, determined by the number of π-electrons participating in the conjugation network. Full control of the number of π-electrons implies full control of the aromatic or antiaromatic character of the molecule. To achieve this, knowledge about the electron delocalization pathways within a molecule is essential. Experimentally, these pathways are not easily accessible. Therefore theoretical studies are necessary to complement experiments that aim at systematically synthesizing porphyrinoids with distinct properties. A computational approach to achieve this is the gauge including magnetically induced current density method (GIMIC). GIMIC is a independent program that is used for the calculation of magnetically induced current densities using London orbitals.[1, 2] Numerical integration of the current flow around molecular rings and along selected chemical bonds can be used for determining current pathways and the degree of aromaticity of various molecules according to the magnetic criterion.[3,4] In my talk I am going to present the GIMIC method and briefly discuss the idea behind it. I will illustrate what type of information is accessible through current density studies. Some very recent results on carbaporphyrinoids will be highlighted showing the potential and the challenges of the method.[5,6]

Open-ended recursive calculation of single residues of response functions for perturbation-dependent basis sets.

We present theory, implementation, and applications of a recursive scheme for the calculation of single residues of response functions that can treat perturbations that affect the basis set. This scheme enables the calculation of nonlinear light absorption properties to arbitrary order for other perturbations than an electric field. We apply this scheme for the first treatment of two-photon circular dichroism (TPCD) using London orbitals at the Hartree-Fock level of theory. In general, TPCD calculations suffer from the problem of origin dependence, which has so far been solved by using the velocity gauge for the electric dipole operator. This work now enables comparison of results from London orbital and velocity gauge based TPCD calculations. We find that the results from the two approaches both exhibit strong basis set dependence but that they are very similar with respect to their basis set convergence.

Relativistic theory of nuclear spin-rotation tensor with kinetically balanced rotational London orbitals.

Both kinetically balanced (KB) and kinetically unbalanced (KU) rotational London orbitals (RLO) are proposed to resolve the slow basis set convergence in relativistic calculations of nuclear spin-rotation (NSR) coupling tensors of molecules containing heavy elements [Y. Xiao and W. Liu, J. Chem. Phys. 138, 134104 (2013)]. While they perform rather similarly, the KB-RLO Ansatz is clearly preferred as it ensures the correct nonrelativistic limit even with a finite basis. Moreover, it gives rise to the same "direct relativistic mapping" between nuclear magnetic resonance shielding and NSR coupling tensors as that without using the London orbitals [Y. Xiao, Y. Zhang, and W. Liu, J. Chem. Theory Comput. 10, 600 (2014)].

Magnetizability and rotational g tensors for density fitted local second-order Møller-Plesset perturbation theory using gauge-including atomic orbitals.

In this paper, we present theory and implementation of an efficient program for calculating magnetizabilities and rotational g tensors of closed-shell molecules at the level of local second-order Møller-Plesset perturbation theory (MP2) using London orbitals. Density fitting is employed to factorize the electron repulsion integrals with ordinary Gaussians as fitting functions. The presented program for the calculation of magnetizabilities and rotational g tensors is based on a previous implementation of NMR shielding tensors reported by S. Loibl and M. Schütz [J. Chem. Phys. 137, 084107 (2012)]. Extensive test calculations show (i) that the errors introduced by density fitting are negligible, and (ii) that the errors of the local approximation are still rather small, although larger than for nuclear magnetic resonance (NMR) shielding tensors. Electron correlation effects for magnetizabilities are tiny for most of the molecules considered here. MP2 appears to overestimate the correlation contribution of magnetizabilities such that it does not constitute an improvement over Hartree-Fock (when comparing to higher-order methods like CCSD(T)). For rotational g tensors the situation is different and MP2 provides a significant improvement in accuracy over Hartree-Fock. The computational performance of the new program was tested for two extended systems, the larger comprising about 2200 basis functions. It turns out that a magnetizability (or rotational g tensor) calculation takes about 1.5 times longer than a corresponding NMR shielding tensor calculation.

On the origin independence of the Verdet tensor†

The condition for invariance under a translation of the coordinate system of the Verdet tensor and the Verdet constant, calculated via quantum chemical methods using gaugeless basis sets, is expressed by a vanishing sum rule involving a third-rank polar tensor. The sum rule is, in principle, satisfied only in the ideal case of optimal variational electronic wavefunctions. In general, it is not fulfilled in non-variational calculations and variational calculations allowing for the algebraic approximation, but it can be satisfied for reasons of molecular symmetry. Group-theoretical procedures have been used to determine (i) the total number of non-vanishing components and (ii) the unique components of both the polar tensor appearing in the sum rule and the axial Verdet tensor, for a series of symmetry groups. Test calculations at the random-phase approximation level of accuracy for water, hydrogen peroxide and ammonia molecules, using basis sets of increasing quality, show a smooth convergence to zero of the sum rule. Verdet tensor components calculated for the same molecules converge to limit values, estimated via large basis sets of gaugeless Gaussian functions and London orbitals.

Gauge-origin independent calculations of Jones birefringence

We present the first gauge-origin independent formulation of Jones birefringence at the Hartree-Fock level of theory. Gauge-origin independence is achieved through the use of London atomic orbitals. The implementation is based on a recently proposed atomic orbital-based response theory formulation that allows for the use of both time- and perturbation-dependent basis sets [Thorvaldsen, Ruud, Kristensen, Jo̸rgensen, and Coriani, J. Chem. Phys. 129, 214108 (2008)]. We present the detailed expressions for the response functions entering the Jones birefringence when London atomic orbitals are used. The implementation is tested on a set of polar and dipolar molecules at the Hartree-Fock level of theory. It is demonstrated that London orbitals lead to much improved basis-set convergence, and that the use of small, conventional basis sets may lead to the wrong sign for the calculated birefringence. For large basis sets, London orbitals and conventional basis sets converge to the same results.

Energy weighted maximum overlap in magnetic fields applications to electron spin resonance problems

Features of the energy weighted maximum overlap (ewmo) model in the presence of a homogeneous magnetic field are analyzed with the view of determining g tensors. London orbitals are used to ensure translational invariance of the results. Several radicals are studied and satisfactory agreement is found between calculated and observed g tensors. These results and previously published ones are used to discuss the possible structure of certain chlorine and bromine oxide radicals.

Theoretical approaches to the calculation of Raman optical activity spectra

In this article, we will give a brief account of the different approaches that have been presented in the literature for calculating Raman optical activity (ROA) spectra by ab initio methods. We will also outline the general structure of a self-consistent-field-based approach for analytic calculations of ROA spectra, including also contributions from London orbitals. The use of London orbitals ensures that the relevant ROA parameters are gauge origin independent. We will also give an outlook on the future of ab initio calculations of Raman optical activity spectra.

Analytical calculations of frequency-dependent hypermagnetizabilities and Cotton–Mouton constants using London atomic orbitals

We present the first gauge-origin-independent, frequency-dependent calculations of the hypermagnetizability anisotropy, which determines the temperature-independent contribution to magnetic-field-induced linear birefringence, the so-called Cotton-Mouton effect. A density-matrix-based scheme for analytical calculations of frequency-dependent molecular properties for self-consistent field models has recently been developed, which is also valid with frequency- and field-dependent basis sets. Applying this scheme to Hartree-Fock wave functions and using London atomic orbitals in order to obtain gauge-origin-independent results, we have calculated the hypermagnetizability anisotropy. Our results show that the use of London orbitals leads to somewhat better basis-set convergence for the hypermagnetizability compared to conventional basis sets and that London orbitals are mandatory in order to obtain reliable magnetizability anisotropies.