Fock Matrix Elements Research Articles

A modular, composite framework for the utilization of reduced-scaling Coulomb and exchange construction algorithms: Design and implementation.

Multiple algorithms exist for calculating Coulomb (J) or exchange (K) contributions to Fock-like matrices, and it is beneficial to develop a framework that allows the seamless integration and combination of different J and K construction algorithms. In Psi4, we have implemented the "CompositeJK" formalism for this purpose. CompositeJK allows for the combination of any J and K construction algorithms for any quantum chemistry method formulated in terms of J-like or K-like matrices (including, but not limited to, Hartree-Fock and density functional theory) in a highly modular and intuitive fashion, which is simple to utilize for both developers and users. Using the CompositeJK framework, Psi4 was interfaced to the sn-LinK implementation in the GauXC library, adding the first instance of noncommercial graphics processing unit (GPU) support for the construction of Fock matrix elements to Psi4. On systems with hundreds of atoms, the interface to the CPU sn-LinK implementation displays a higher performance than all the alternative JK construction methods available in Psi4, with up to x2.8 speedups compared to existing Psi4JK implementations. The GPU sn-LinK implementation, harnessing the power of GPUs, improves the observed performance gains to up to x7.0.

Highly Efficient and Accurate Computation of Multiple Orbital Spaces Spanning Fock Matrix Elements on Central and Graphics Processing Units for Application in F12 Theory.

We employ our recently published highly efficient seminumerical exchange (sn-LinK) [Laqua, H.; Thompson, T. H.; Kussmann, J.; Ochsenfeld, C. J. Chem. Theory Comput. 2020, 16, 1456-1468] and integral-direct resolution of the identity Coulomb (RI-J) [Kussmann, J.; Laqua, H.; Ochsenfeld, C. J. Chem. Theory Comput. 2021, 17, 1512-1521] methods to significantly accelerate the computation of the demanding multiple orbital spaces spanning Fock matrix elements present in R12/F12 theory on central and graphics processing units. The errors introduced by RI-J and sn-LinK into the RI-MP2-F12 energy are thoroughly assessed for a variety of basis sets and integration grids. We find that these numerical errors are always below "chemical accuracy" (∼1 mH) even for the coarsest settings and can easily be reduced below 1 μH by employing only moderately large integration grids and RI-J basis sets. Since the number of basis functions of the multiple orbital spaces is notably larger compared with conventional Hartree-Fock theory, the efficiency gains from the superior basis scaling of RI-J and sn-LinK (O(Nbas2) instead of O(Nbas4) for both) are even more significant, with maximum speedup factors of 37 000 for RI-J and 4500 for sn-LinK. In total, the multiple orbital spaces spanning Fock matrix evaluation of the largest tested structure using a triple-ζ F12 basis set (5058 AO basis functions, 9267 CABS basis functions) is accelerated over 1575× using CPUs and over 4155× employing GPUs.

Multi-Headed Symmetrical Superpositions of Coherent States

Based on N different coherent states with equal weights and phase-space rotation symmetry, we introduce N-headed incoherent superposition states (NHICSSs) and N-headed coherent superposition states (NHCSSs). These N coherent states are associated with N-order roots of the same complex number. We study and compare properties of NHICSSs and NHCSSs, including average photon number, Mandel Q parameter, quadrature squeezing, Fock matrix elements and Wigner function. Among all these states, only 2HCSS (i.e., Schrodinger cat state) presents quadrature-squeezing effect. Our theoretical results can be used as a reference for researchers in this field.

Amplified thermal state: Properties and decoherence

In this paper, we introduce the amplified thermal state (ATS) by operating [Formula: see text] on the thermal state (TS). Here, [Formula: see text] is the amplification factor and [Formula: see text] is the photon number operator. We study its properties, such as light intensity, signal-to-noise ratio (SNR), Fock matrix elements and Wigner function. In addition, we study its decoherence in photon-loss channel by analyzing evolution of all above properties. All considered properties are derived analytically and simulated numerically. Compared with the original TS, the amplification can enhance light intensity and SNR, remain the mixed character, and exhibit non-Gaussianity. While the decoherence will weaken light intensity and SNR, remain the mixed character, and return to Gaussian state.

Scalable Electron Correlation Methods. 8. Explicitly Correlated Open-Shell Coupled-Cluster with Pair Natural Orbitals PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12.

We present explicitly correlated open-shell pair natural orbital local coupled-cluster methods, PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12. The methods are extensions of our previously reported PNO-R/UCCSD methods (J. Chem. Theory Comput., 2020, 16, 3135-3151, https://pubs.acs.org/doi/10.1021/acs.jctc.0c00192) with additions of explicit correlation and perturbative triples corrections. The explicit correlation treatment follows the spin-orbital CCSD-F12b theory using Ansatz 3*A, which is found to yield comparable or better basis set convergence than the more rigorous Ansatz 3C in computed ionization potentials and reaction energies using double- to quaduple-ζ basis sets. The perturbative triples correction is adapted from the spin-orbital (T) theory to use triples natural orbitals (TNOs). To address the coupling due to off-diagonal Fock matrix elements, the local triples amplitudes are iteratively solved using small domains of TNOs, and a semicanonical (T0) domain correction with larger domains is applied to reduce the domain errors. The performance of the methods is demonstrated through benchmark calculations on ionization potentials, radical stabilization energies, reaction energies of fragmentations and rearrangements in radical cations, and spin-state energy differences of iron complexes. For a few test sets where canonical calculations are feasible, PNO-RCCSD(T)-F12 results agree with the canonical ones to within 0.4 kcal mol-1, and this maximum error is reduced to below 0.2 kcal mol-1 when large local domains are used. For larger systems, results using different thresholds for the local approximations are compared to demonstrate that 1 kcal mol-1 level of accuracy can be achieved using our default settings. For a couple of difficult cases, it is demonstrated that the errors from individual approximations are only a fraction of 1 kcal mol-1, and the overall accuracy of the method does not rely on error compensations. In contrast to canonical calculations, the use of spin-orbitals does not lead to a significant increase of computational time and memory usage in the most expensive steps of PNO-R/UCCSD(T)-F12 calculations. The only exception is the iterative solution of the (T) amplitudes, which can be avoided without significant errors by using a perturbative treatment of the off-diagonal coupling, known as (T1) approximation. For most systems, even the semicanonical approximation (T0) leads only to small errors in relative energies. Our program is well parallelized and capable of computing accurate correlation energies for molecules with 100-200 atoms using augmented triple-ζ basis sets in less than a day of elapsed time on a small computer cluster.

Changing Fock matrix elements of two-mode squeezed vacuum state by employing three conditional operations in one-sided lossy channel

This paper focuses on changing Fock matrix elements of two-mode squeezed vacuum state (TMSVS) by employing three conditional operations in one-sided lossy channel. These three conditional operations include one-photon replacement (OPR), one-photon substraction (OPS) and one-photon addition (OPA). Indeed, three conditional quantum states have been generated from the original TMSVS. Using the characteristic function (CF) representation of quantum density operator, we derive the analytical expressions of their Fock matrix elements, which depend on the interaction parameters, including the squeezing parameter of the input TMSVS, the loss factor and the transmissivity of the variable beam splitter. For convenience of discussion, we only give the Fock matrices in the subspace span for these two-mode states. Obviously, the TMSVS only has the populations in and in such subspace. By comparing the generated states with the TMSVS, we find that: (1) the generated state after OPR will remain the populations in and , and add the populations in and (2) the generated state after OPS will lost the populations in and , but add the populations in and (3) the generated state after OPA will remain the population only in and add the population in .

Linear scaling perturbative triples correction approximations for open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory [DLPNO-CCSD(T0/T)

The coupled cluster method with single-, double-, and perturbative triple excitations [CCSD(T)] is considered to be one of the most reliable quantum chemistry theories. However, the steep scaling of CCSD(T) has limited its application to small or medium-sized systems for a long time. In our previous work, the linear scaling domain based local pair natural orbital CCSD variant (DLPNO-CCSD) has been developed for closed-shell and open-shell. However, it is known from extensive benchmark studies that triple-excitation contributions are important to reach chemical accuracy. In the present work, two linear scaling (T) approximations for open-shell DLPNO-CCSD are implemented and compared: (a) an algorithm based on the semicanonical approximation, in which off-diagonal Fock matrix elements in the occupied space are neglected [referred to as DLPNO-(T0)]; and (b) an improved algorithm in which the triples amplitudes are computed iteratively [referred to as DLPNO-(T)]. This work is based on the previous open-shell DLPNO-CCSD algorithm [M. Saitow et al., J. Chem. Phys. 146, 164105 (2017)] as well as the iterative (T) correction for closed-shell systems [Y. Guo et al., J. Chem. Phys. 148, 011101 (2018)]. Our results show that the new open-shell perturbative corrections, DLPNO-(T0/T), can predict accurate absolute and relative correlation energies relative to the canonical reference calculations with the same basis set. The absolute energies from DLPNO-(T) are significantly more accurate than those of DLPNO-(T0). The additional computational effort of DLPNO-(T) relative to DLPNO-(T0) is a factor of 4 on average. We report calculations on systems with more than 4000 basis functions.

Simulating realistic non-Gaussian state preparation

We consider conditional photonic non-Gaussian state preparation using multimode Gaussian states and photon-number-resolving detectors in the presence of photon loss. While simulation of such state preparation is often computationally challenging, we show that obtaining the required multimode Gaussian state Fock matrix elements can be reduced to the computation of matrix functions known as loop hafnians, and develop a tailored algorithm for their calculation that is faster than previously known methods. As an example of its utility, we use our algorithm to explore the loss parameter space for three specific non-Gaussian state preparation schemes: Fock state heralding, cat state heralding, and weak cubic-phase state heralding. We confirm that these schemes are fragile with respect to photon loss, yet find that there are regions in the loss parameter space that are potentially accessible in an experimental setting which correspond to heralded states with non-zero non-Gaussianity.

Transferability in Machine Learning for Electronic Structure via the Molecular Orbital Basis.

We present a machine learning (ML) method for predicting electronic structure correlation energies using Hartree-Fock input. The total correlation energy is expressed in terms of individual and pair contributions from occupied molecular orbitals, and Gaussian process regression is used to predict these contributions from a feature set that is based on molecular orbital properties, such as Fock, Coulomb, and exchange matrix elements. With the aim of maximizing transferability across chemical systems and compactness of the feature set, we avoid the usual specification of ML features in terms of atom- or geometry-specific information, such atom/element-types, bond-types, or local molecular structure. ML predictions of MP2 and CCSD energies are presented for a range of systems, demonstrating that the method maintains accuracy while providing transferability both within and across chemical families; this includes predictions for molecules with atom-types and elements that are not included in the training set. The method holds promise both in its current form and as a proof-of-principle for the use of ML in the design of generalized density-matrix functionals.

Scalable Electron Correlation Methods. 5. Parallel Perturbative Triples Correction for Explicitly Correlated Local Coupled Cluster with Pair Natural Orbitals.

A well-parallelized perturbative triples correction implementation for the pair natural orbital based coupled cluster method PNO-LCCSD(T)-F12 is presented. A composite approach is adopted in addressing the coupling due to off-diagonal Fock matrix elements, in which the local triples amplitudes are iteratively solved using small domains of triples natural orbitals, and a semicanonical (T0) domain correction with larger domains is applied to reduce the domain errors. This treatment adds only about 20% to the computational cost of (T0) calculations with large domains, and the errors due to the use of small domains in the iterations are very small. In addition, a two-step triple list selection method is applied: First, an initial triple list is generated using LCCSD-F12 pair energy criteria, and the (T0) triples energies are computed using small domains. Second, this list is reduced by neglecting triples with small energy contributions, and the final calculation with large domains is only carried out for the reduced list. The cost of the (T) calculation scales asymptotically linear with the molecular size and shows excellent parallelization efficiency up to hundreds of CPU cores. The convergence of the (T) contribution to the relative energies of large molecular systems is carefully tested, and for most of the cases the results obtained with our default thresholds agree within ∼0.5 kcal mol-1 with those computed with very tight thresholds. For all tested molecular systems where canonical calculations are still feasible, the PNO-LCCSD(T)-F12 relative energies also agree within 0.5 kcal mol-1 with the canonical CCSD(T)-F12 results using the same F12 treatment. The (T) calculation generally takes 3-5 times the cost of the preceding PNO-LCCSD-F12 calculation, primarily due to the large number of triples required in obtaining accurate relative energies. We find that for large molecular systems the triple selection criteria used in previous local triples methods are insufficient, and a much larger number of triples are required than it was assumed so far. Still, using a commodity computer cluster, the PNO-LCCSD(T)-F12 calculation of molecules with ∼100 atoms using augmented triple-ζ basis sets can be carried out in a few hours of elapsed time.

Perturbative triples correction for local pair natural orbital based explicitly correlated CCSD(F12*) using Laplace transformation techniques.

We present an implementation of pair natural orbital coupled cluster singles and doubles with perturbative triples, PNO-CCSD(T), which avoids the quasi-canonical triples approximation (T0) where couplings due to off-diagonal Fock matrix elements are neglected. A numerical Laplace transformation of the canonical expression for the perturbative (T) triples correction is used to avoid an I/O and storage bottleneck for the triples amplitudes. Results for a test set of reaction energies show that only very few Laplace grid points are needed to obtain converged energy differences and that PNO-CCSD(T) is a more robust approximation than PNO-CCSD(T0) with a reduced mean absolute deviation from canonical CCSD(T) results. We combine the PNO-based (T) triples correction with the explicitly correlated PNO-CCSD(F12*) method and investigate the use of specialized F12-PNOs in the conventional triples correction. We find that no significant additional errors are introduced and that PNO-CCSD(F12*)(T) can be applied in a black box manner.

Accelerating ab initio molecular dynamics simulations by linear prediction methods

Acceleration of ab initio molecular dynamics (AIMD) simulations can be reliably achieved by extrapolation of electronic data from previous timesteps. Existing techniques utilize polynomial least-squares regression to fit previous steps’ Fock or density matrix elements. In this work, the recursive Burg ‘linear prediction’ technique is shown to be a viable alternative to polynomial regression, and the extrapolation-predicted Fock matrix elements were three orders of magnitude closer to converged elements. Accelerations of 1.8–3.4× were observed in test systems, and in all cases, linear prediction outperformed polynomial extrapolation. Importantly, these accelerations were achieved without reducing the MD integration timestep.

H₂ Adsorbed Site-to-Site Electronic Delocalization within IRMOF-1: Understanding Non-Negligible Interactions at High Pressure.

Isoreticular metal organic frameworks (IRMOFs) have shown high uptake capabilities for storage of H2 (11.5 wt % at 77 K and 170 bar). A significant literature has employed fragment models and a single adsorbed H2 to identify adsorption sites within IRMOFs, as well as the necessary adsorbate–adsorbent interactions needed to reach sufficient adsorption enthalpy for practical usage, however at high pressures it remains to be seen if H2···H2 intermolecular interactions may influence the energetics. This study focuses upon IRMOF-1 (also known as MOF-5), and examines the individual H2 stabilization energies at different sites using Möller–Plesset perturbation theory and density functional theory alongside chemical models that consist of isolated fragment models and a cubic super cell cluster consisting of both the face- and edge-cube’s of IRMOF-1. Optimization of twenty stable configurations of singly adsorbed H2 in the super-cell cluster is observed to be essential to obtain energy ordering of the five primary sites consistent with experiment and prior benchmark calculations (α >> β > γ > δ ≈ ε). To examine site-to-site interactions that may occur in the high-pressure regime, 64 co-adsorbed H2 within a super-cell cluster have been studied (a theoretical maximum of all adsorption sites, 14 wt %). There, delocalization and/or charge transfer of electrons is observed from the σ orbitals of the H2 bound at the γ positions into the σ* orbitals of H2 bound at the α sites leads to stabilization of the interaction of H2 at the γ, by 1.4 kJ/mol, respectively (using M06-2X/LANL2DZ). This effect has been confirmed to be charge transfer, and not a manifestation of enhanced dispersion at high loading, through natural bond order (NBO) analysis and by comparisons of the square of off-diagonal NBO Fock matrix elements for both density functionals that account for dispersion interactions and Hartree–Fock calculations that ignore dispersion.

Fourier Space Uncoupled Hartree–Fock Polarizabilities of One-Dimensionally Periodic Systems. Polyethylene and Polysilane Revisited

Abstract A method for the computation of electronic energies of stereoregular polymers is reviewed, using a formulation that makes use of Fourier-representation techniques and the Ewald procedure for accelerating the convergence of lattice sums. That method is in the present work extended to include the computation of electric polarization at an “uncoupled” approximation at the level of second-order perturbation theory based on Hartree–Fock wave functions, using the procedure of Blount and of Genkin and Mednis, as applied to polymers by Barbier, Delhalle, and André. The extension requires computation of the derivatives of Fock matrix elements with respect to the Bloch-wave parameter k, and an effficient numerical procedure for evaluating these derivatives is described here. The computational procedures are incorporated in the authors' ft-1d program, and the new features of that program are validated by reexamining the band structures of polyethylene and polysilane. The results are consistent with the older work on these systems, but exhibit more computational efficiency and greater achievable accuracy.

Two-component multi-configurational second-order perturbation theory with Kramers restricted complete active space self-consistent field reference function and spin-orbit relativistic effective core potential.

We report the formulation and implementation of KRCASPT2, a two-component multi-configurational second-order perturbation theory based on Kramers restricted complete active space self-consistent field (KRCASSCF) reference function, in the framework of the spin-orbit relativistic effective core potential. The zeroth-order Hamiltonian is defined as the sum of nondiagonal one-electron operators with generalized two-component Fock matrix elements as scalar factors. The Kramers symmetry within the zeroth-order Hamiltonian is maintained via the use of a state-averaged density, allowing a consistent treatment of degenerate states. The explicit expressions are derived for the matrix elements of the zeroth-order Hamiltonian as well as for the perturbation vector. The use of a fully variational reference function and nondiagonal operators in relativistic multi-configurational perturbation theory is reported for the first time. A series of initial calculations are performed on the ionization potential and excitation energies of the atoms of the 6p-block; the results display a significant improvement over those from KRCASSCF, showing a closer agreement with experimental results. Accurate atomic properties of the superheavy elements of the 7p-block are also presented, and the electronic structures of the low-lying excited states are compared with those of their lighter homologues.

A theoretical investigation of the structure of 2-nitropyridine-N-oxide and the dependency of the NO2 torsional motion on the applied wavefunction and basis set.

HF, B3LYP, and MP2 wave functions in combination with Pople 6-31, 6-311 augmented with polarization functions on all atoms and Dunning double- and triple-zeta basis sets have been employed to investigate the structures and torsional potential function of the nitro group in 2-nitropyridine-N-oxide (2-NPO) and a variety of its fluorinated derivatives. The augmentation of the basis sets with diffuse functions showed a marked effect on the profile and barriers of the NO2 torsional potential. Depending on the applied model chemistry, the heterocyclic ring in some 2-NPOs has proved to be non-planar. The non-planarity of the ring was characterized by Cremer–Pople puckering amplitude Q. The disruption of the ring planarity in some NPOs was accounted for the distinctive reactivity and impact sensitivity of these heterocycles. Consistently, the NBO and the AIM analyses furnished clear evidence for the accentuated weakness of the C–NO2 bond and provided evidence for the electronic interplay between the NO2 group, the fluorine substituent and the heterocyclic ring. Deletion of all off-diagonal Fock-matrix elements (NOSTAR) to separate hyperconjugative stabilizing interactions from steric interactions was used. The effect of nitration and fluorination on the aromaticity of the studied 2-NPOs was investigated by using the NICS descriptors NICS(1) and NICS(1)zz. These NICS indices have shown that the fluorination in para position to the nitro group exhibits the highest degree of aromaticity within the fluorinated 2-NPOs.Electronic supplementary materialThe online version of this article (doi:10.1007/s11224-012-0198-5) contains supplementary material, which is available to authorized users.

Construction of the two-electron contribution to the Fock matrix by numerical integration

A novel method to numerically calculate the Fock matrix is presented. The Coulomb operator is re-expressed as an integral identity, which is discretized. The discretization of the auxiliary t dimension separates the x, y, and z dependencies transforming the two-electron Coulomb integrals of Gaussian-type orbitals (GTO) to a linear sum of products of two-dimensional integrals. The s-type integrals are calculated analytically and integrals of the higher angular-momentum functions are obtained using recursion formulae. The contributions to the two-body Coulomb integrals obtained for each discrete t value can be evaluated independently. The two-body Fock matrix elements can be integrated numerically, using common sets of quadrature points and weights. The aim is to calculate Fock matrices of enough accuracy for electronic structure calculations. Preliminary calculations indicate that it is possible to achieve an overall accuracy of at least 10−12 E h using the numerical approach.

Study of C [dbnd]O⋯H [sbnd]N Hydrogen bond interactions in amyloid beta (A β): A DFT study of the electric field gradient and CS tensors and NBO analysis

A systematic computational density functional theory (DFT) study was carried out in amyloid-beta (Aβ) structure to investigate the hydrogen bonding interactions (HBs) effects on Chemical shielding (CS) and electric field gradient (EFG) tensors in site of hydrogen, oxygen, nitrogen and carbon nuclei. The results indicate that these tensors are more sensitive to the HB interactions in dimer model than those of trimer model. The calculated 15N and 13C in CS tensors in trimer model are in good agreement with experimental data. Natural bonding orbital analysis, NBO, indicate a reasonable correlation between 14N quadrupole coupling constant ( C Q ) and strength of Fock matrix elements ( F ij ). Moreover, the quantum chemical calculations indicated that the intermolecular the HB interactions play an essential role in variation of the relative orientation of CS and EFG tensors of nitrogen, hydrogen and oxygen nuclei in the molecular frame axes. Finally, 17O C Q and η Q show great dependency to backbone conformation, specifically dihedral angles ψ (psi).

Using multipole point charge distributions to provide the electrostatic potential in the variational explicit polarization (X-Pol) potential

The equations defining the variational explicit polarization (X-Pol) potential introduced in earlier work are modified in the present work so that multipole point charge distributions are used instead of Mulliken charges to polarize the monomers that comprise the system. In addition, when computing the electrostatic interaction between a monomer whose molecular orbitals are being optimized and a monomer whose electron density is being used to polarize the first monomer, the electron densities of both monomers are represented by atom-centered multipole point charge distributions. In the original formulation of the variational X-Pol potential, the continuous electron density of the monomer being optimized interacts with external Mulliken charges, but this corresponds to the monopole truncation in a multipole expansion scheme in the computation of the Fock matrix elements of the given monomer. The formulation of the variational X-Pol potential introduced in this work (which we are calling the "multipole variational X-Pol potential") represents the electron density of the monomer whose wave function is being variationally optimized in the same way that it represents the electron densities of external monomers when computing the Coulomb interactions between them.

Theoretical study of N–H· · ·O hydrogen bonding properties and cooperativity effects in linear acetamide clusters

We investigated geometry, energy, \({\nu_{{\text{N--H}}}}\) harmonic frequencies, 14N nuclear quadrupole coupling tensors, and \({n_{\rm O}\to \sigma _{{\text{N--H}}}^\ast}\) charge transfer properties of (acetamide) n clusters, with n = 1 − 7, by means of second-order Moller-Plesset perturbation theory (MP2) and DFT method. Dependency of dimer stabilization energies and equilibrium geometries on various levels of theory was examined. B3LYP/6-311++G** calculations revealed that for acetamide clusters, the average hydrogen-bonding energy per monomer increases from −26.85 kJ mol−1 in dimer to −35.12 kJ mol−1 in heptamer; i.e., 31% cooperativity enhancement. The n-dependent trend of \({\nu_{{\text{N--H}}}\,{and}\,^{14}}\) N nuclear quadrupole coupling values were reasonably correlated with cooperative effects in \({r_{{\text{N--H}}}}\) bond distance. It was also found that intermolecular \({n_{\rm O}\to \sigma_{{\text{N--H}}}^\ast}\) charge transfer plays a key role in cooperative changes of geometry, binding energy, \({\nu_{{\text{N--H}}}}\) harmonic frequencies, and 14N electric field gradient tensors of acetamide clusters. There is a good linear correlation between 14N quadrupole coupling constants, C Q (14N), and the strength of Fock matrix elements (F ij ). Regarding the \({n_{\rm O}\to \sigma_{{\text{N--H}}}^\ast}\) interaction, the capability of the acetamide clusters for electron localization, at the N–H· · ·O bond critical point, depends on the cluster size and thereby leads to cooperative changes in the N–H· · ·O length and strength, N–H stretching frequencies, and 14N quadrupole coupling tensors.