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The Importance of Zeroth-Order Approximations in Molecular Quantum Mechanics

David Stück

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

The work herein is concerned with developing computational models to understand molecules. The underlying theme of this research is the reassessment of zeroth-order approximations for higher-level methods. For second-order Moller-Plesset theory (MP2), qualitative failures of the Hartree-Fock orbitals in the form of spin contamination can lead to catastrophic errors in the second order energies. By working with orbitals optimized in the presence of correlations, orbital-optimized MP2 can fix the spin contamination problem that plague radicals, aromatics, and transition metal complexes. In path integral Monte Carlo for vibrational energies, the zeroth-order propagator is typically chosen to be the most general possible, the free particle propagator; we chose to be informed by the molecular structure we have already attained and apply a propagator based on the harmonic modes of the molecule, improving sampling efficiency and our Trotter approximation.

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Convergent hierarchies of theories for calculating many-body vibrational ground and excited-state wave functions, such as M{o}ller-Plesset perturbation theory or coupled cluster theory, tend to rely on matrix-algebraic manipulations of large, high-dimensional arrays of anharmonic force constants, tasks which require large amounts of computer storage space and which are very difficult to implement in a parallel-scalable fashion. On the other hand, existing quantum Monte Carlo (QMC) methods for vibrational wave functions tend to lack robust techniques for obtaining excited-state energies, especially for large systems. By exploiting analytical identities for matrix elements of position operators in a harmonic oscillator basis, we have developed stochastic implementations of the size-extensive vibrational self-consistent field (MC-XVSCF) and size-extensive vibrational M{o}ller-Plesset second-order perturbation (MC-XVMP2) theories which do not require storing the potential energy surface (PES). The programmable equations of MC-XVSCF and MC-XVMP2 take the form of a small number of high-dimensional integrals evaluated using Metropolis Monte Carlo techniques. The associated integrands require independent evaluations of only the value, not the derivatives, of the PES at many points, a task which is trivial to parallelize. However, unlike existing vibrational QMC methods, MC-XVSCF and MC-XVMP2 can calculate anharmonic frequencies directly, rather than as a small difference between two noisy total energies, and do not require user-selected coordinates or nodal surfaces. MC-XVSCF and MC-XVMP2 can also directly sample the PES in a given approximation without analytical or grid-based approximations, enabling us to quantify the errors induced by such approximations.

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Quantum Mechanics for Quantum Chemistry

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Abstract ................................................................................................. 359 4.1 Introduction .................................................................................. 359 4.2 Feynman’s Path Integral Quantum Formalism ............................ 364 4.2.1 Construction of Path Integral ........................................... 364 4.2.2 Propagator’s Equation ...................................................... 370 4.2.3 Recovering Wave Function’s Equation ............................ 373 4.3 Path Integrals for Basic Matter’s Structures ................................ 377 4.3.1 General Path Integral’s Properties ................................... 377 4.3.2 Path Integral for the Free Particle .................................... 384 4.3.3 Path Integral for Motion as the Harmonic Oscillator ...... 387 4.3.4 Path Integral Representation for the Bohr’s Atom ........... 394 4.3.5 Path Integral for Motion in the Quantum Well ................ 398 4.4 Density Matrix Approach Linking Path Integral Formalism ....... 402 4.4.1 On Mono-, Many-, and Reduced-Electronic Density Matrices .............................................................. 402 4.4.2 Canonical Density, Bloch Equation, and the Need of Path Integral ....................................................... 412 4.5 Roots of Self-Consistent Methods in Quantum Chemistry .......... 419 4.5.1 Molecular Orbital Approach ............................................ 420 4.5.2 Roothaan Approach .......................................................... 423 4.5.3 Introducing Semi-Empirical Approximations .................. 428 4.5.3.1 NDO Methods ................................................... 429 4.5.3.2 NDDO Methods ................................................ 432 4.5.4 Ab initio Methods: The Hartree-Fock Approach ............. 436 4.5.4.1 Hartree-Fock Orbital Energy ............................ 438 4.5.4.2 About Correlation Energy ................................. 442 4.5.4.3 Koopman’s Orbital Theorem with Hartree-Fock Picture ......................................... 444 4.5.4.4 Chemical Reactivity Indices in Orbital Energy Representation ...................................... 451 4.5.4.5 Testing Koopmans Theorem by Chemical Harness Reactivity Index .................. 458 4.6 Density Functional Theory: Observable Quantum Chemistry ..... 468 4.6.1 Hohenberg-Kohn theorems .............................................. 470 4.6.2 Optimized Energy-Electronegativity Connection ............ 475 4.6.3 Popular Energetic Density Functionals ............................ 480 4.6.3.1 Density Functionals of Kinetic Energy ............. 481 4.6.3.2 Density Functionals of Exchange Energy ......... 484 4.6.3.3 Density Functionals of Correlation Energy ...... 490 4.6.3.4 Density Functionals of Exchange-Correlation Energy ........................... 496 4.7 Observable Quantum Chemistry: Extending Heisenberg’s Uncertainty ............................................................. 502 4.7.1 Periodic Path Integrals ..................................................... 502 4.7.1.1 Effective Partition Function .............................. 502 4.7.1.2 Periodic Quantum Paths .................................... 505 4.7.2 Heisenberg Uncertainty Reloaded ................................... 508 4.7.3 Extending Heisenberg Uncertainty ...................................511 4.7.3.1 Averaging Quantum Fluctuations ......................511 4.7.3.2 Observed Quantum Evolution .......................... 514 4.7.3.3 Free Quantum Evolution ................................... 515 4.7.3.4 Free vs. Observed Quantum Evolution ............. 518 4.7.3.5 Spectral-IQ Method .......................................... 523 4.7.3.6 Spectral-IQ Results on Silica Sol-Gel-Based Mesosystems ..................................................... 527 4.8 Conclusion ................................................................................... 533 Keywords .............................................................................................. 535 References ............................................................................................. 536 Author’s Main References ........................................................... 536 Specific References ...................................................................... 537 Further Readings .......................................................................... 549 ABSTRACT Basing on the first principles of Quantum mechanics as exposed in the previous chapters and sections, special chapters of quantum theory are here unfolded in order to further extend and caching the quantum information from free to observed evolution within the matter systems with constraints (boundaries). As such, the Feynman path integral formalism is firstly exposed and then applied to atomic, quantum barrier and quantum harmonically vibration, followed by density matrix approach, opening the Hartree-Fock and Density Functional pictures of many-electronic systems, with a worthy perspective of electronic occupancies via Koopmans theorem, while ending with a further generalization of the Heisenberg observability and of its first application to mesosystems.

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Post-Hartree-Fock methods are wavefunction-based quantum chemistry approaches that explicitly account for the electron correlation beyond the mean-field approximation, allowing accurate determination of electron orbitals and energies in a given system. However, their computational cost scales as the fourth power or higher with respect to the number of atoms, severely limiting their practical applications. This paper proposes an efficient GPU implementation of the Møller-Plesset method, a post-Hartree-Fock approach based on perturbation theory. We focus on the computation of second- and thirdorder perturbation energies (MP2 and MP3) for closed-shell systems, which incorporate electron correlation on top of the Hartree-Fock energy. To accelerate the extensive contraction operations arising in the Møller-Plesset method, our implementation employs a formulation that maximizes the use of highly optimized matrix multiplication routines. Furthermore, we effectively utilize computational capabilities of GPU architecture through a hierarchical reduction using warp-level primitives and shared memory, as well as contiguous coalesced access to matrix elements. Performance evaluation on an NVIDIA A100 GPU shows up to 255.5 and 71.1 times speedups for the MP2 correlation energy computation, compared with the widely used quantum chemistry software package PySCF and its GPU plugin GPU4PySCF, respectively. In the MP3 calculations, our DGEMM-based implementation achieves a maximum speedup of 96.2 times over PySCF.

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Performance of the fixed-node diffusion quantum Monte Carlo method (FN-DMC) with a single Slater-Jastrow trial wavefunction using natural orbitals (NOs) from MP2, CCSD, CCSD(T), and CASSCF as well as density functional theory orbitals on atomization energies of the molecules in the Gaussian-2 set is investigated in this work. The effects of spin contamination and pseudopotentials (PPs) are also studied. Our results show that the DMC energy with NOs from MP2 or CCSD(T) is the lowest on average, whereas that from CASSCF is only lower than the DMC energy using Hartree-Fock orbitals. Atomization energies are generally underestimated with DMC, and mean absolute deviations of DMC atomization energies are about 2.8 kcal/mol with NOs from MP2 or CCSD(T) and about 2.7 kcal/mol with NOs from CASSCF or B3LYP orbitals. The better performance of the latter two orbitals is due to a more effective error cancellation. The accuracy of the present FN-DMC is similar to CCSD(T)/aug-cc-pVQZ on atomization energies. In addition, the error of DMC atomization energies tends to be larger for molecules with multiple bonds. DMC energies of the open-shell systems with spin-restricted orbitals are generally higher than those with spin-unrestricted orbitals if spin contamination is not serious and the atomization energies are improved to some extent. Furthermore, our results indicate that the error of the PPs is rather small at the CCSD(T) level, but could be more pronounced in DMC calculations and DMC atomization energies are improved with some newly developed PPs.

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Springer eBooks•Jan 1, 2011•0 citations
The work herein is concerned with developing computational models to understand molecules. The underlying theme of this research is the reassessment of zeroth-order approximations for higher-level methods. For second-order Moller-Plesset theory (MP2), qualitative failures of the Hartree-Fock orbitals in the form of spin contamination can lead to catastrophic errors in the second order energies. By working with orbitals optimized in the presence of correlations, orbital-optimized MP2 can fix the spin contamination problem that plague radicals, aromatics, and transition metal complexes. In path integral Monte Carlo for vibrational energies, the zeroth-order propagator is typically chosen to be the most general possible, the free particle propagator; we chose to be informed by the molecular structure we have already attained and apply a propagator based on the harmonic modes of the molecule, improving sampling efficiency and our Trotter approximation.
Convergent hierarchies of theories for calculating many-body vibrational ground and excited-state wave functions, such as M{o}ller-Plesset perturbation theory or coupled cluster theory, tend to rely on matrix-algebraic manipulations of large, high-dimensional arrays of anharmonic force constants, tasks which require large amounts of computer storage space and which are very difficult to implement in a parallel-scalable fashion. On the other hand, existing quantum Monte Carlo (QMC) methods for vibrational wave functions tend to lack robust techniques for obtaining excited-state energies, especially for large systems. By exploiting analytical identities for matrix elements of position operators in a harmonic oscillator basis, we have developed stochastic implementations of the size-extensive vibrational self-consistent field (MC-XVSCF) and size-extensive vibrational M{o}ller-Plesset second-order perturbation (MC-XVMP2) theories which do not require storing the potential energy surface (PES). The programmable equations of MC-XVSCF and MC-XVMP2 take the form of a small number of high-dimensional integrals evaluated using Metropolis Monte Carlo techniques. The associated integrands require independent evaluations of only the value, not the derivatives, of the PES at many points, a task which is trivial to parallelize. However, unlike existing vibrational QMC methods, MC-XVSCF and MC-XVMP2 can calculate anharmonic frequencies directly, rather than as a small difference between two noisy total energies, and do not require user-selected coordinates or nodal surfaces. MC-XVSCF and MC-XVMP2 can also directly sample the PES in a given approximation without analytical or grid-based approximations, enabling us to quantify the errors induced by such approximations.
Abstract ................................................................................................. 359 4.1 Introduction .................................................................................. 359 4.2 Feynman’s Path Integral Quantum Formalism ............................ 364 4.2.1 Construction of Path Integral ........................................... 364 4.2.2 Propagator’s Equation ...................................................... 370 4.2.3 Recovering Wave Function’s Equation ............................ 373 4.3 Path Integrals for Basic Matter’s Structures ................................ 377 4.3.1 General Path Integral’s Properties ................................... 377 4.3.2 Path Integral for the Free Particle .................................... 384 4.3.3 Path Integral for Motion as the Harmonic Oscillator ...... 387 4.3.4 Path Integral Representation for the Bohr’s Atom ........... 394 4.3.5 Path Integral for Motion in the Quantum Well ................ 398 4.4 Density Matrix Approach Linking Path Integral Formalism ....... 402 4.4.1 On Mono-, Many-, and Reduced-Electronic Density Matrices .............................................................. 402 4.4.2 Canonical Density, Bloch Equation, and the Need of Path Integral ....................................................... 412 4.5 Roots of Self-Consistent Methods in Quantum Chemistry .......... 419 4.5.1 Molecular Orbital Approach ............................................ 420 4.5.2 Roothaan Approach .......................................................... 423 4.5.3 Introducing Semi-Empirical Approximations .................. 428 4.5.3.1 NDO Methods ................................................... 429 4.5.3.2 NDDO Methods ................................................ 432 4.5.4 Ab initio Methods: The Hartree-Fock Approach ............. 436 4.5.4.1 Hartree-Fock Orbital Energy ............................ 438 4.5.4.2 About Correlation Energy ................................. 442 4.5.4.3 Koopman’s Orbital Theorem with Hartree-Fock Picture ......................................... 444 4.5.4.4 Chemical Reactivity Indices in Orbital Energy Representation ...................................... 451 4.5.4.5 Testing Koopmans Theorem by Chemical Harness Reactivity Index .................. 458 4.6 Density Functional Theory: Observable Quantum Chemistry ..... 468 4.6.1 Hohenberg-Kohn theorems .............................................. 470 4.6.2 Optimized Energy-Electronegativity Connection ............ 475 4.6.3 Popular Energetic Density Functionals ............................ 480 4.6.3.1 Density Functionals of Kinetic Energy ............. 481 4.6.3.2 Density Functionals of Exchange Energy ......... 484 4.6.3.3 Density Functionals of Correlation Energy ...... 490 4.6.3.4 Density Functionals of Exchange-Correlation Energy ........................... 496 4.7 Observable Quantum Chemistry: Extending Heisenberg’s Uncertainty ............................................................. 502 4.7.1 Periodic Path Integrals ..................................................... 502 4.7.1.1 Effective Partition Function .............................. 502 4.7.1.2 Periodic Quantum Paths .................................... 505 4.7.2 Heisenberg Uncertainty Reloaded ................................... 508 4.7.3 Extending Heisenberg Uncertainty ...................................511 4.7.3.1 Averaging Quantum Fluctuations ......................511 4.7.3.2 Observed Quantum Evolution .......................... 514 4.7.3.3 Free Quantum Evolution ................................... 515 4.7.3.4 Free vs. Observed Quantum Evolution ............. 518 4.7.3.5 Spectral-IQ Method .......................................... 523 4.7.3.6 Spectral-IQ Results on Silica Sol-Gel-Based Mesosystems ..................................................... 527 4.8 Conclusion ................................................................................... 533 Keywords .............................................................................................. 535 References ............................................................................................. 536 Author’s Main References ........................................................... 536 Specific References ...................................................................... 537 Further Readings .......................................................................... 549 ABSTRACT Basing on the first principles of Quantum mechanics as exposed in the previous chapters and sections, special chapters of quantum theory are here unfolded in order to further extend and caching the quantum information from free to observed evolution within the matter systems with constraints (boundaries). As such, the Feynman path integral formalism is firstly exposed and then applied to atomic, quantum barrier and quantum harmonically vibration, followed by density matrix approach, opening the Hartree-Fock and Density Functional pictures of many-electronic systems, with a worthy perspective of electronic occupancies via Koopmans theorem, while ending with a further generalization of the Heisenberg observability and of its first application to mesosystems.
Post-Hartree-Fock methods are wavefunction-based quantum chemistry approaches that explicitly account for the electron correlation beyond the mean-field approximation, allowing accurate determination of electron orbitals and energies in a given system. However, their computational cost scales as the fourth power or higher with respect to the number of atoms, severely limiting their practical applications. This paper proposes an efficient GPU implementation of the Møller-Plesset method, a post-Hartree-Fock approach based on perturbation theory. We focus on the computation of second- and thirdorder perturbation energies (MP2 and MP3) for closed-shell systems, which incorporate electron correlation on top of the Hartree-Fock energy. To accelerate the extensive contraction operations arising in the Møller-Plesset method, our implementation employs a formulation that maximizes the use of highly optimized matrix multiplication routines. Furthermore, we effectively utilize computational capabilities of GPU architecture through a hierarchical reduction using warp-level primitives and shared memory, as well as contiguous coalesced access to matrix elements. Performance evaluation on an NVIDIA A100 GPU shows up to 255.5 and 71.1 times speedups for the MP2 correlation energy computation, compared with the widely used quantum chemistry software package PySCF and its GPU plugin GPU4PySCF, respectively. In the MP3 calculations, our DGEMM-based implementation achieves a maximum speedup of 96.2 times over PySCF.
Performance of the fixed-node diffusion quantum Monte Carlo method (FN-DMC) with a single Slater-Jastrow trial wavefunction using natural orbitals (NOs) from MP2, CCSD, CCSD(T), and CASSCF as well as density functional theory orbitals on atomization energies of the molecules in the Gaussian-2 set is investigated in this work. The effects of spin contamination and pseudopotentials (PPs) are also studied. Our results show that the DMC energy with NOs from MP2 or CCSD(T) is the lowest on average, whereas that from CASSCF is only lower than the DMC energy using Hartree-Fock orbitals. Atomization energies are generally underestimated with DMC, and mean absolute deviations of DMC atomization energies are about 2.8 kcal/mol with NOs from MP2 or CCSD(T) and about 2.7 kcal/mol with NOs from CASSCF or B3LYP orbitals. The better performance of the latter two orbitals is due to a more effective error cancellation. The accuracy of the present FN-DMC is similar to CCSD(T)/aug-cc-pVQZ on atomization energies. In addition, the error of DMC atomization energies tends to be larger for molecules with multiple bonds. DMC energies of the open-shell systems with spin-restricted orbitals are generally higher than those with spin-unrestricted orbitals if spin contamination is not serious and the atomization energies are improved to some extent. Furthermore, our results indicate that the error of the PPs is rather small at the CCSD(T) level, but could be more pronounced in DMC calculations and DMC atomization energies are improved with some newly developed PPs.