Numerical evaluation of orientation averages and its application to molecular physics

In molecular physics, it is often necessary to average over the orientation of molecules when calculating observables, in particular when modeling experiments in the liquid or gas phase. Evaluated in terms of Euler angles, this is closely related to integration over two- or three-dimensional unit spheres, a common problem discussed in numerical analysis. The computational cost of the integration depends significantly on the quadrature method, making the selection of an appropriate method crucial for the feasibility of simulations. After reviewing several classes of spherical quadrature methods in terms of their efficiency and error distribution, we derive guidelines for choosing the best quadrature method for orientation averages and illustrate these with three examples from chiral molecule physics. While Gauss quadratures allow for achieving numerically exact integration for a wide range of applications, other methods offer advantages in specific circumstances. Our guidelines can also be applied to higher-dimensional spherical domains and other geometries. We also present a Python package providing a flexible interface to a variety of quadrature methods.

A neural network-based four-body potential energy surface for parahydrogen

We present an isotropic ab initio (para-H2)4 four-body interaction potential energy surface (PES). The electronic structure calculations are performed at the correlated coupled-cluster theory level, with single, double, and perturbative triple excitations. They use an atom-centered augmented correlation-consistent double zeta basis set, supplemented by a (3s3p2d) midbond function. We use a multilayer perceptron to construct the PES. We apply a rescaling transformation to the output energies during training to improve the prediction of weaker energies in the sample data. At long distances, the interaction energies are adjusted to match the empirically derived four-body dispersion interaction. The four-body interaction energy at short intermolecular separations is net repulsive. The use of this four-body PES, in combination with a first principles pair potential for para-H2 [J. Chem. Phys. 119, 12551 (2015)] and an isotropic ab initio three-body potential for para-H2 [J. Chem. Phys. 156, 044301 (2022)], is expected to provide closer agreement with experimental results.

Exact quantum dynamics developments for floppy molecular systems and complexes

Molecular rotation, vibration, internal rotation, isomerization, tunneling, intermolecular dynamics of weakly and strongly interacting systems, intra-to-inter-molecular energy transfer, hindered rotation and hindered translation over surfaces are important types of molecular motions. Their fundamentally correct and detailed description can be obtained by solving the nuclear Schrödinger equation on a potential energy surface. Many of the chemically interesting processes involve quantum nuclear motions which are 'delocalized' over multiple potential energy wells. These 'large-amplitude' motions in addition to the high dimensionality of the vibrational problem represent challenges to the current (ro)vibrational methodology. A review of the quantum nuclear motion methodology is provided, current bottlenecks of solving the nuclear Schrödinger equation are identified, and solution strategies are reviewed. Technical details, computational results, and analysis of these results in terms of limiting models and spectroscopically relevant concepts are highlighted for selected numerical examples.

Fitting potential energy and induced dipole surfaces of the van der Waals complex CH4-N2 using non-product quadrature grids

We present an extensive study of the five-dimensional potential energy and induced dipole surfaces of the CH₄-N₂ complex assuming rigid-rotor approximation. Within the supermolecular approach, ab initio calculations of the interaction energies and dipoles were carried out at the CCSD(T)-F12 and CCSD(T) levels of theory using the correlation-consistent aug-cc-pVTZ basis set, respectively. Both potential energy and induced dipole surfaces inherit the symmetry of the molecular system and transform under the A₁⁺ and A₂⁺ irreducible representations of the molecular symmetry group G₄₈, respectively. One can take advantage of the symmetry when fitting the surfaces; first, when constructing angular basis functions and second, when selecting the grid points. The approach to the construction of scalar and vectorial basis functions exploiting the eigenfunction method [Q. Chen, J. Ping and F. Wang, Group Representation Theory for Physicists, World Scientific, 2nd edn, 2002] is developed. We explore the use of Sobolev-type quadrature grids as building blocks of robust quadrature rules adapted to the symmetry of the molecular system. Temperature variations of the cross second virial coefficient and first classical spectral moments of the rototranslational collision-induced band were derived. A reasonable agreement between calculated values and experimental data was found attesting to the high quality of constructed surfaces.

Toward breaking the curse of dimensionality in (ro)vibrational computations of molecular systems with multiple large-amplitude motions

Methodological progress is reported in the challenging direction of a black-box-type variational solution of the (ro)vibrational Schrödinger equation applicable to floppy, polyatomic systems with multiple large-amplitude motions. This progress is achieved through the combination of (i) the numerical kinetic-energy operator (KEO) approach of Mátyus et al. [J. Chem. Phys. 130, 134112 (2009)] and (ii) the Smolyak nonproduct grid method of Avila and Carrington, Jr. [J. Chem. Phys. 131, 174103 (2009)]. The numerical representation of the KEO makes it possible to choose internal coordinates and a body-fixed frame best suited for the molecular system. The Smolyak scheme reduces the size of the direct-product grid representation by orders of magnitude, while retaining some of the useful features of it. As a result, multidimensional (ro)vibrational states are computed with system-adapted coordinates, a compact basis- and grid-representation, and an iterative eigensolver. Details of the methodological developments and the first numerical applications are presented for the CH₄·Ar complex treated in full (12D) vibrational dimensionality.

Nuclear magnetic relaxation by the dipolar EMOR mechanism: Three-spin systems

In aqueous systems with immobilized macromolecules, including biological tissue, the longitudinal spin relaxation of water protons is primarily induced by exchange-mediated orientational randomization (EMOR) of intra- and intermolecular magnetic dipole-dipole couplings. Starting from the stochastic Liouville equation, we have developed a non-perturbative theory that can describe relaxation by the dipolar EMOR mechanism over the full range of exchange rates, dipole couplings, and Larmor frequencies. Here, we implement the general dipolar EMOR theory for a macromolecule-bound three-spin system, where one, two, or all three spins exchange with the bulk solution phase. In contrast to the previously studied two-spin system with a single dipole coupling, there are now three dipole couplings, so relaxation is affected by distinct correlations as well as by self-correlations. Moreover, relaxation can now couple the magnetizations with three-spin modes and, in the presence of a static dipole coupling, with two-spin modes. As a result of this complexity, three secondary dispersion steps with different physical origins can appear in the longitudinal relaxation dispersion profile, in addition to the primary dispersion step at the Larmor frequency matching the exchange rate. Furthermore, and in contrast to the two-spin system, longitudinal relaxation can be significantly affected by chemical shifts and by the odd-valued ("imaginary") part of the spectral density function. We anticipate that the detailed studies of two-spin and three-spin systems that have now been completed will provide the foundation for developing an approximate multi-spin dipolar EMOR theory sufficiently accurate and computationally efficient to allow quantitative molecular-level interpretation of frequency-dependent water-proton longitudinal relaxation data from biophysical model systems and soft biological tissue.

Nuclear magnetic relaxation induced by exchange-mediated orientational randomization: longitudinal relaxation dispersion for a dipole-coupled spin-1/2 pair

In complex biological or colloidal samples, magnetic relaxation dispersion (MRD) experiments using the field-cycling technique can characterize molecular motions on time scales ranging from nanoseconds to microseconds, provided that a rigorous theory of nuclear spin relaxation is available. In gels, cross-linked proteins, and biological tissues, where an immobilized macromolecular component coexists with a mobile solvent phase, nuclear spins residing in solvent (or cosolvent) species relax predominantly via exchange-mediated orientational randomization (EMOR) of anisotropic nuclear (electric quadrupole or magnetic dipole) couplings. The physical or chemical exchange processes that dominate the MRD typically occur on a time scale of microseconds or longer, where the conventional perturbation theory of spin relaxation breaks down. There is thus a need for a more general relaxation theory. Such a theory, based on the stochastic Liouville equation (SLE) for the EMOR mechanism, is available for a single quadrupolar spin I = 1. Here, we present the corresponding theory for a dipole-coupled spin-1/2 pair. To our knowledge, this is the first treatment of dipolar MRD outside the motional-narrowing regime. Based on an analytical solution of the spatial part of the SLE, we show how the integral longitudinal relaxation rate can be computed efficiently. Both like and unlike spins, with selective or non-selective excitation, are treated. For the experimentally important dilute regime, where only a small fraction of the spin pairs are immobilized, we obtain simple analytical expressions for the auto-relaxation and cross-relaxation rates which generalize the well-known Solomon equations. These generalized results will be useful in biophysical studies, e.g., of intermittent protein dynamics. In addition, they represent a first step towards a rigorous theory of water (1)H relaxation in biological tissues, which is a prerequisite for unravelling the molecular basis of soft-tissue contrast in clinical magnetic resonance imaging.

N₂reduction and hydrogenation to ammonia by a molecular iron-potassium complex

The most common catalyst in the Haber-Bosch process for the hydrogenation of dinitrogen (N(2)) to ammonia (NH(3)) is an iron surface promoted with potassium cations (K(+)), but soluble iron complexes have neither reduced the N-N bond of N(2) to nitride (N(3-)) nor produced large amounts of NH(3) from N(2). We report a molecular iron complex that reacts with N(2) and a potassium reductant to give a complex with two nitrides, which are bound to iron and potassium cations. The product has a Fe(3)N(2) core, implying that three iron atoms cooperate to break the N-N triple bond through a six-electron reduction. The nitride complex reacts with acid and with H(2) to give substantial yields of N(2)-derived ammonia. These reactions, although not yet catalytic, give structural and spectroscopic insight into N(2) cleavage and N-H bond-forming reactions of iron.

Assessment of DFT and DFT-D for Potential Energy Surfaces of Rare Gas Trimers-Implementation and Analysis of Functionals and Extrapolation Procedures

Given the recent developments in methodology associated with the accurate computation of molecular systems with weak interactions, it is of particular interest to revisit systems that are notoriously challenging for determining reliable potential energy surface (PES) descriptions. Additionally, challenges associated with carrying out complete basis set extrapolation procedures and treatment of basis set superposition error (BSSE) are of importance in these descriptions. In this work, investigation into the ability to accurately predict the potential energy surfaces of the main Rg3 molecules (Rg = He, Ne, Ar) is made across a range of wave function types and large basis sets, including the use of several established extrapolation procedures and counterpoise corrections. Wave function types span most classes of density functional types, including the newest DFT-D schemes, and are benchmarked against high accuracy CCSD(T)/CBS methodology. Study of such systems is valuable, as they serve as simple models for many complex properties, most importantly n-body weak interaction energies.

Full-potential multiple scattering theory with space-filling cells for bound and continuum states

We present a rigorous derivation of a real-space full-potential multiple scattering theory (FP-MST) that is free from the drawbacks that up to now have impaired its development (in particular the need to expand cell shape functions in spherical harmonics and rectangular matrices), valid both for continuum and bound states, under conditions for space partitioning that are not excessively restrictive and easily implemented. In this connection we give a new scheme to generate local basis functions for the truncated potential cells that is simple, fast, efficient, valid for any shape of the cell and reduces to the minimum the number of spherical harmonics in the expansion of the scattering wavefunction. The method also avoids the need for saturating 'internal sums' due to the re-expansion of the spherical Hankel functions around another point in space (usually another cell center). Thus this approach provides a straightforward extension of MST in the muffin-tin (MT) approximation, with only one truncation parameter given by the classical relation l(max) = kR(b), where k is the electron wavevector (either in the excited or ground state of the system under consideration) and R(b) is the radius of the bounding sphere of the scattering cell. Moreover, the scattering path operator of the theory can be found in terms of an absolutely convergent procedure in the l(max) --> ∞ limit. Consequently, this feature provides a firm ground for the use of FP-MST as a viable method for electronic structure calculations and makes possible the computation of x-ray spectroscopies, notably photo-electron diffraction, absorption and anomalous scattering among others, with the ease and versatility of the corresponding MT theory. Some numerical applications of the theory are presented, both for continuum and bound states.

Full-potential multiple scattering for core electron spectroscopies

We present a rigorous derivation of a real space full-potential multiple-scattering theory (FP-MST), valid both for continuum and bound states, that is free from the drawbacks that up to now have impaired its development, in particular the need to use cell shape functions and rectangular matrices. In this connection we give a new scheme to generate local basis functions for the truncated potential cells that is simple, fast, efficient, valid for any shape of the cell and reduces to the minimum the number of spherical harmonics in the expansion of the scattering wavefunction. This approach provides a straightforward extension of MST in the muffin-tin (MT) approximation, with only one truncation parameter given by the classical relation l(max) = kR(b), where k is the photo-electron wavevector and R(b) the radius of the bounding sphere of the scattering cell. Some numerical applications of the theory are presented, both for continuum and bound states.

Efficient orientational averaging by the extension of Lebedev grids via regularized octahedral symmetry expansion

Gaussian spherical quadrature methods in the guise of the Lebedev sampling grids are highly efficient for some orientational ("powder") averaging problems in solid state NMR. However, their applicability is currently restricted, as the sets of orientations are derived analytically and because they are not well adapted to simulate the broad peakshapes encountered, for example, in the NMR on static powders or on half-integer quadrupolar spins subject to second order quadrupolar interactions under magic-angle spinning conditions. We remedy these problems by (i) introducing the recursive procedure regularized octahedral symmetry expansion (ROSE), to which any existing Lebedev set may be subjected. Each recursive step gives a 9-fold enlarged set of orientations. (ii) We demonstrate that ROSE-expanded grids, in conjunction with spectral interpolation, is well suited for calculating broad peakshapes. These advances combine into the apparently most efficient general-purpose two-angle orientational averaging technique proposed to date for solid state NMR applications.

Calculating multidimensional discrete variable representations from cubature formulas

Finding multidimensional nondirect product discrete variable representations (DVRs) of Hamiltonian operators is one of the long standing challenges in computational quantum mechanics. The concept of a "DVR set" was introduced as a general framework for treating this problem by R. G. Littlejohn, M. Cargo, T. Carrington, Jr., K. A. Mitchell, and B. Poirier (J. Chem. Phys. 2002, 116, 8691). We present a general solution of the problem of calculating multidimensional DVR sets whose points are those of a known cubature formula. As an illustration, we calculate several new nondirect product cubature DVRs on the plane and on the sphere with up to 110 points. We also discuss simple and potentially very useful finite basis representations (FBRs), based on general (nonproduct) cubatures. Connections are drawn to a novel view on cubature presented by I. Degani, J. Schiff, and D. J. Tannor (Num. Math. 2005, 101, 479), in which commuting extensions of coordinate matrices play a central role. Our construction of DVR sets answers a problem left unresolved in the latter paper, namely, the problem of interpreting as function spaces the vector spaces on which commuting extensions act.

Finite basis representations with nondirect product basis functions having structure similar to that of spherical harmonics

The currently most efficient finite basis representation (FBR) method [Corey et al., in Numerical Grid Methods and Their Applications to Schrodinger Equation, NATO ASI Series C, edited by C. Cerjan (Kluwer Academic, New York, 1993), Vol. 412, p. 1; Bramley et al., J. Chem. Phys. 100, 6175 (1994)] designed specifically to deal with nondirect product bases of structures phi(n) (l)(s)f(l)(u), chi(m) (l)(t)phi(n) (l)(s)f(l)(u), etc., employs very special l-independent grids and results in a symmetric FBR. While highly efficient, this method is not general enough. For instance, it cannot deal with nondirect product bases of the above structure efficiently if the functions phi(n) (l)(s) [and/or chi(m) (l)(t)] are discrete variable representation (DVR) functions of the infinite type. The optimal-generalized FBR(DVR) method [V. Szalay, J. Chem. Phys. 105, 6940 (1996)] is designed to deal with general, i.e., direct and/or nondirect product, bases and grids. This robust method, however, is too general, and its direct application can result in inefficient computer codes [Czako et al., J. Chem. Phys. 122, 024101 (2005)]. It is shown here how the optimal-generalized FBR method can be simplified in the case of nondirect product bases of structures phi(n) (l)(s)f(l)(u), chi(m) (l)(t)phi(n) (l)(s)f(l)(u), etc. As a result the commonly used symmetric FBR is recovered and simplified nonsymmetric FBRs utilizing very special l-dependent grids are obtained. The nonsymmetric FBRs are more general than the symmetric FBR in that they can be employed efficiently even when the functions phi(n) (l)(s) [and/or chi(m) (l)(t)] are DVR functions of the infinite type. Arithmetic operation counts and a simple numerical example presented show unambiguously that setting up the Hamiltonian matrix requires significantly less computer time when using one of the proposed nonsymmetric FBRs than that in the symmetric FBR. Therefore, application of this nonsymmetric FBR is more efficient than that of the symmetric FBR when one wants to diagonalize the Hamiltonian matrix either by a direct or via a basis-set contraction method. Enormous decrease of computer time can be achieved, with respect to a direct application of the optimal-generalized FBR, by employing one of the simplified nonsymmetric FBRs as is demonstrated in noniterative calculations of the low-lying vibrational energy levels of the H3+ molecular ion. The arithmetic operation counts of the Hamiltonian matrix vector products and the properties of a recently developed diagonalization method [Andreozzi et al., J. Phys. A Math. Gen. 35, L61 (2002)] suggest that the nonsymmetric FBR applied along with this particular diagonalization method is suitable to large scale iterative calculations. Whether or not the nonsymmetric FBR is competitive with the symmetric FBR in large-scale iterative calculations still has to be investigated numerically.