Dual-Basis Analytic Gradients. 1. Self-Consistent Field Theory

Analytic gradients for local density fitting Hartree-Fock and Kohn-Sham methods

We present analytic gradients for local density fitting Hartree-Fock (HF) and hybrid Kohn-Sham (KS) density functional methods. Due to the non-variational nature of the local fitting algorithm, the method of Lagrange multipliers is used to avoid the solution of the coupled perturbed HF and KS equations. We propose efficient algorithms for the solution of the arising Z-vector equations and the gradient calculation that preserve the third-order scaling and low memory requirement of the original local fitting algorithm. In order to demonstrate the speed and accuracy of our implementation, gradient calculations and geometry optimizations are presented for various molecular systems. Our results show that significant speedups can be achieved compared to conventional density fitting calculations without sacrificing accuracy.

Adaptive-Partitioning Multilayer Dynamics Simulations: 1. On-the-Fly Switch between Two Quantum Levels of Theory

We propose to generalize the previously developed two-layer permuted adaptive-partitioning quantum-mechanics/molecular-mechanics (QM/MM), which reclassifies atoms as QM or MM on-the-fly in dynamics simulations, to multilayer adaptive-partitioning algorithms that enable multiple levels of theory. In this work, we formulate two new algorithms that smoothly interpolate the energy between two QM (Q1 and Q2) levels of theory. The first "permuted adaptive-partitioning" scheme is based on the weighted many-body expansion of the potential, as in the adaptive-partitioning QM/MM. Unconventional and potentially more efficient, the second "interpolated adaptive-partitioning" method employs alchemical QM calculations with Q1/Q2-mixed basis sets, Fock matrices, and overlap matrices. To our knowledge, this is the first time that such alchemical calculations are performed in QM, although they are routinely done in MM. Test calculations on water-cluster models show that both new algorithms indeed yield smooth energy curves when water molecules shift between Q1 and Q2.

Speeding up Hartree-Fock and Kohn-Sham calculations with first-order corrections

Several approaches are presented to improve the efficiency of Hartree-Fock and Kohn-Sham self-consistent field (SCF) calculations relying on a simple first-order energy correction reminiscent of the scheme used in dual-basis SCF methods. The basic idea is to perform an initial SCF calculation computing approximate Fock-matrices and, in the final iteration step, to use a more complete Fock-matrix builder together with the energy correction to diminish the error. The approximation is tested for conventional and local density fitting (DF) SCF approaches combining various auxiliary basis sets, fitting metrics, and Fock-matrix construction algorithms in the initial and final iterations as well as for seminumerical SCF methods combining integration grids of different qualities. We also report the implementation of the occupied orbital resolution of identity exchange construction algorithm with local DF approximations. Benchmark calculations are presented for total energies, reaction energies, and molecular geometries. Our results show that speedups of up to 80% can be expected utilizing the new approaches without significant loss of accuracy.

A Linear-Scaling Method for Noncovalent Interactions: An Efficient Combination of Absolutely Localized Molecular Orbitals and a Local Random Phase Approximation Approach

A novel method for the accurate and efficient calculation of interaction energies in weakly bound complexes composed of a large number of molecules is presented. The new ALMO+RPAd method circumvents the prohibitive scaling of coupled cluster singles and doubles while still providing similar accuracy across a diverse range of weakly bound chemical systems. Linear-scaling procedures for the Fock build are given utilizing absolutely localized molecular orbitals (ALMOs), resulting in the a priori exclusion of basis set superposition errors. A bespoke data structure and algorithm using density fitting are described, leading to linear scaling for the storage and computation of the two-electron integrals. Electron correlation is included through a new, linear-scaling pairwise local random phase approximation approach, including exchange interactions, and decomposed into purely dispersive excitations (RPAxd). Collectively, these allow meaningful decomposition of the interaction energy into physically distinct contributions: electrostatic, polarization, charge transfer, and dispersion. Comparison with symmetry-adapted perturbation theory shows good qualitative agreement. Tests on various dimers and the S66 benchmark set demonstrate results within 0.5 kcal mol^-1 of coupled cluster singles and doubles results. On a large cluster of water molecules, we achieve calculations involving over 3500 orbital and 12,000 auxiliary basis functions in under 10 min on a single CPU core.

Computation of Hydration Free Energies Using the Multiple Environment Single System Quantum Mechanical/Molecular Mechanical Method

A recently developed MESS-E-QM/MM method (multiple-environment single-system quantum mechanical molecular/mechanical calculations with a Roothaan-step extrapolation) is applied to the computation of hydration free energies for the blind SAMPL4 test set and for 12 small molecules. First, free energy simulations are performed with a classical molecular mechanics force field using fixed-geometry solute molecules and explicit TIP3P solvent, and then the non-Boltzmann-Bennett method is employed to compute the QM/MM correction (QM/MM-NBB) to the molecular mechanical hydration free energies. For the SAMPL4 set, MESS-E-QM/MM-NBB corrections to the hydration free energy can be obtained 2 or 3 orders of magnitude faster than fully converged QM/MM-NBB corrections, and, on average, the hydration free energies predicted with MESS-E-QM/MM-NBB fall within 0.10-0.20 kcal/mol of full-converged QM/MM-NBB results. Out of five density functionals (BLYP, B3LYP, PBE0, M06-2X, and ωB97X-D), the BLYP functional is found to be most compatible with the TIP3P solvent model and yields the most accurate hydration free energies against experimental values for solute molecules included in this study.

Ab initio investigation of the aqueous solvation of the nitrate ion

The turning point between surface and interior solvation of NO₃⁻ is predicted to lie around a cluster size of (H₂O)₆₄.

Multiple environment single system quantum mechanical/molecular mechanical (MESS-QM/MM) calculations. 1. Estimation of polarization energies

In combined quantum mechanical/molecular mechanical (QM/MM) free energy calculations, it is often advantageous to have a frozen geometry for the quantum mechanical (QM) region. For such multiple-environment single-system (MESS) cases, two schemes are proposed here for estimating the polarization energy: the first scheme, termed MESS-E, involves a Roothaan step extrapolation of the self-consistent field (SCF) energy; whereas the other scheme, termed MESS-H, employs a Newton–Raphson correction using an approximate inverse electronic Hessian of the QM region (which is constructed only once). Both schemes are extremely efficient, because the expensive Fock updates and SCF iterations in standard QM/MM calculations are completely avoided at each configuration. They produce reasonably accurate QM/MM polarization energies: MESS-E can predict the polarization energy within 0.25 kcal/mol in terms of the mean signed error for two of our test cases, solvated methanol and solvated β-alanine, using the M06-2X or ωB97X-D functionals; MESS-H can reproduce the polarization energy within 0.2 kcal/mol for these two cases and for the oxyluciferin–luciferase complex, if the approximate inverse electronic Hessians are constructed with sufficient accuracy.

At what chain length do unbranched alkanes prefer folded conformations?

Short unbranched alkanes are known to prefer linear conformations, whereas long unbranched alkanes are folded. It is not known with certainty at what chain length the linear conformation is no longer the global minimum. To clarify this point, we use ab initio and density functional methods to compute the relative energies of the linear and hairpin alkane conformers for increasing chain lengths. Extensive electronic structure calculations are performed to obtain optimized geometries, harmonic frequencies, and accurate single point energies for the selected alkane conformers from octane through octadecane. Benchmark CCSD(T)/cc-pVTZ single point calculations are performed for chains through tetradecane, whereas approximate methods are required for the longer chains up to octadecane. Using frozen natural orbitals to unambiguously truncate the virtual orbital space, we are able to compute composite CCSD FNO(T) single point energies for all the chain lengths. This approximate composite method has significant computational savings compared to full CCSD(T) while retaining ∼0.15 kcal/mol accuracy compared to the benchmark results. More approximate dual-basis resolution-of-the-identity double-hybrid DFT calculations are also performed and shown to have reasonable 0.2-0.4 kcal/mol errors compared with our benchmark values. After including contributions from temperature dependent internal energy shifts, we find the preference for folded conformations to lie between hexadecane and octadecane, in excellent agreement with recent experiments [ Lüttschwager , N. O. ; Wassermann , T. N. ; Mata , R. A. ; Suhm , M. A. Angew. Chem. Int. Ed. 2013 , 52 , 463 ].

Behavior of density functionals with respect to basis set. VI. Truncation of the correlation consistent basis sets

Systematic truncation of the correlation consistent basis sets has been investigated in first and second row atoms and molecules to better understand basis set requirements for density functional theory, particularly the need for high angular momentum functions, as well as to understand possible computational cost savings that could be achieved by using reduced basis sets. The truncation scheme employed follows that recently introduced for ab initio methods [B. Mintz et al., J. Chem. Phys. 121, 5629 (2004)]. Properties examined in the current study include geometries, ionization potentials, electron affinities, and dissociation energies. In general, this investigation shows that a degree of truncation of higher angular momentum functions is possible with limited impact upon energetic properties, and does result in useful CPU time savings. However, not all properties investigated have the same level of dependence upon high angular momentum functions, and, thus, careful selection of truncated basis sets should be made.