Combining the Complete Active Space Self-Consistent Field Method and the Full Configuration Interaction Quantum Monte Carlo within a Super-CI Framework, with Application to Challenging Metal-Porphyrins.

A novel stochastic Complete Active Space Self-Consistent Field (CASSCF) method has been developed and implemented in the Molcas software package. A two-step procedure is used, in which the CAS configuration interaction secular equations are solved stochastically with the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) approach, while orbital rotations are performed using an approximated form of the Super-CI method. This new method does not suffer from the strong combinatorial limitations of standard MCSCF implementations using direct schemes and can handle active spaces well in excess of those accessible to traditional CASSCF approaches. The density matrix formulation of the Super-CI method makes this step independent of the size of the CI expansion, depending exclusively on one- and two-body density matrices with indices restricted to the relatively small number of active orbitals. No sigma vectors need to be stored in memory for the FCIQMC eigensolver--a substantial gain in comparison to implementations using the Davidson method, which require three or more vectors of the size of the CI expansion. Further, no orbital Hessian is computed, circumventing limitations on basis set expansions. Like the parent FCIQMC method, the present technique is scalable on massively parallel architectures. We present in this report the method and its application to the free-base porphyrin, Mg(II) porphyrin, and Fe(II) porphyrin. In the present study, active spaces up to 32 electrons and 29 orbitals in orbital expansions containing up to 916 contracted functions are treated with modest computational resources. Results are quite promising even without accounting for the correlation outside the active space. The systems here presented clearly demonstrate that large CASSCF calculations are possible via FCIQMC-CASSCF without limitations on basis set size.

[1]  Kimihiko Hirao,et al.  THEORETICAL STUDY OF THE ELECTRONIC GROUND STATE OF IRON(II) PORPHINE , 1998 .

[2]  B. Lévy.  Molecular MC–SCF calculations , 1970 .

[3]  Roland Lindh,et al.  Main group atoms and dimers studied with a new relativistic ANO basis set , 2004 .

[4]  Ali Alavi,et al.  Unbiased reduced density matrices and electronic properties from full configuration interaction quantum Monte Carlo. , 2014, The Journal of chemical physics.

[5]  Kimihiko Hirao,et al.  THEORETICAL STUDY OF THE ELECTRONIC GROUND STATE OF IRON(II) PORPHINE. II , 1999 .

[6]  D. Hartree.  The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  U. Schollwoeck.  The density-matrix renormalization group , 2004, cond-mat/0409292.

[8]  Ali Alavi,et al.  Communications: Survival of the fittest: accelerating convergence in full configuration-interaction quantum Monte Carlo. , 2010, The Journal of chemical physics.

[9]  H. Nakatsuji,et al.  SAC-CI Study on the Excited and Ionized States of Free-Base Porphin: Rydberg Excited States and Effect of Polarization and Rydberg Functions , 1998 .

[10]  Martin Gouterman,et al.  Spectra of porphyrins , 1961 .

[11]  Björn O. Roos,et al.  Second-order perturbation theory with a complete active space self-consistent field reference function , 1992 .

[12]  Sason Shaik,et al.  Reactivity of high-valent iron-oxo species in enzymes and synthetic reagents: a tale of many states. , 2007, Accounts of chemical research.

[13]  Theoretical determination of the electronic spectrum of free base porphin , 1994 .

[14]  Hiroshi Nakatsuji,et al.  EXCITED AND IONIZED STATES OF FREE BASE PORPHIN STUDIED BY THE SYMMETRY ADAPTED CLUSTER-CONFIGURATION INTERACTION (SAC-CI) METHOD , 1996 .

[15]  C. Reed,et al.  Nuclear magnetic resonance investigation of magnetic and electronic properties of "intermediate spin" ferrous porphyrin complexes. , 1977, Journal of the American Chemical Society.

[16]  Ali Alavi,et al.  A study of electron affinities using the initiator approach to full configuration interaction quantum Monte Carlo. , 2011, The Journal of chemical physics.

[17]  Donald G Truhlar,et al.  Multiconfiguration Pair-Density Functional Theory: A Fully Translated Gradient Approximation and Its Performance for Transition Metal Dimers and the Spectroscopy of Re2Cl8(2-). , 2015, Journal of chemical theory and computation.

[18]  I. Shavitt.  Graph theoretical concepts for the unitary group approach to the many-electron correlation problem , 2009 .

[19]  Sriram Krishnamoorthy,et al.  Active-space completely-renormalized equation-of-motion coupled-cluster formalism: Excited-state studies of green fluorescent protein, free-base porphyrin, and oligoporphyrin dimer. , 2010, The Journal of chemical physics.

[20]  Klaus Ruedenberg,et al.  MCSCF optimization through combined use of natural orbitals and the brillouin–levy–berthier theorem , 1979 .

[21]  B. Roos,et al.  Ground state free base porphin: C2v or D2h symmetry? A theoretical contribution , 1994 .

[22]  Thomas G. Spiro,et al.  Low-Lying Spin States of Iron(II) Porphine , 1998 .

[23]  Ali Alavi,et al.  An excited-state approach within full configuration interaction quantum Monte Carlo. , 2015, The Journal of chemical physics.

[24]  Jeppe Olsen,et al.  Determinant based configuration interaction algorithms for complete and restricted configuration interaction spaces , 1988 .

[25]  Donald G Truhlar,et al.  Multiconfiguration pair-density functional theory: barrier heights and main group and transition metal energetics. , 2015, Journal of chemical theory and computation.

[26]  Rebecca K. Carlson,et al.  Multiconfiguration Pair-Density Functional Theory. , 2014, Journal of chemical theory and computation.

[27]  T. Straatsma,et al.  Multiplet splittings and other properties from density functional theory: an assessment in iron–porphyrin systems , 2005 .

[28]  B. Roos,et al.  Theoretical study of the electronic spectrum of magnesium-porphyrin , 1999 .

[29]  Markus Reiher,et al.  Density matrix renormalization group with efficient dynamical electron correlation through range separation. , 2015, The Journal of chemical physics.

[30]  Ali Alavi,et al.  Semi-stochastic full configuration interaction quantum Monte Carlo: Developments and application. , 2015, The Journal of chemical physics.

[31]  Manuela Merchán,et al.  Interpretation of the electronic absorption spectrum of free base porphin by using multiconfigurational second-order perturbation theory , 1998 .

[32]  Michael Grätzel,et al.  Porphyrin-Sensitized Solar Cells with Cobalt (II/III)–Based Redox Electrolyte Exceed 12 Percent Efficiency , 2011, Science.

[33]  Rodney J. Bartlett,et al.  Coupled-cluster calculations of the electronic excitation spectrum of free base porphin in a polarized basis , 1998 .

[34]  James B. Anderson,et al.  Quantum chemistry by random walk. H 2P, H+3D3h1A′1, H23Σ+u, H41Σ+g, Be 1S , 1976 .

[35]  M. Gouterman,et al.  SELF-CONSISTENT MOLECULAR ORBITAL CALCULATIONS OF PORPHYRIN AND RELATED RING SYSTEMS. , 1965 .

[36]  I. Shavitt.  Matrix element evaluation in the unitary group approach to the electron correlation problem , 1978 .

[37]  James B. Anderson,et al.  A random‐walk simulation of the Schrödinger equation: H+3 , 1975 .

[38]  C. Reed,et al.  Mössbauer effect study of the magnetic properties of S=1 ferrous tetraphenylporphyrin , 1978 .

[39]  U. Nagashima,et al.  Abinitio SCF‐CI calculation on free base porphin and chlorin; theoretical analysis on intensities of the absorption spectra , 1986 .

[40]  Uzi Even,et al.  Isolated ultracold porphyrins in supersonic expansions. III. Free‐base porphine , 1982 .

[41]  Matthias Troyer,et al.  An efficient matrix product operator representation of the quantum chemical Hamiltonian. , 2015, The Journal of chemical physics.

[42]  K. Pierloot,et al.  Multiconfigurational Second-Order Perturbation Theory Restricted Active Space (RASPT2) Studies on Mononuclear First-Row Transition-Metal Systems. , 2011, Journal of chemical theory and computation.

[43]  Giovanni Li Manni,et al.  The generalized active space concept in multiconfigurational self-consistent field methods. , 2011, The Journal of chemical physics.

[44]  S. Scheiner,et al.  Electronic structure and bonding in unligated and ligated FeII porphyrins , 2002 .

[45]  Felipe Zapata,et al.  Molcas 8: New capabilities for multiconfigurational quantum chemical calculations across the periodic table , 2016, J. Comput. Chem..

[46]  K. De Wael,et al.  Comparison between the electrocatalytic properties of different metal ion phthalocyanines and porphyrins towards the oxidation of hydroxide. , 2008, Talanta.

[47]  F. Grein,et al.  Convergence behavior of some multiconfiguration methods , 1976 .

[48]  B. Roos,et al.  A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach , 1980 .

[49]  T. Noro,et al.  Ab initio CI calculations on free-base porphin , 1992 .

[50]  J. Mispelter,et al.  Proton magnetic resonance characterization of the intermediate (S=1) spin state of ferrous porphyrins , 1980 .

[51]  David Dolphin,et al.  Porphyrins XVII. Vapor absorption spectra and redox reactions: Tetraphenylporphins and porphin , 1971 .

[52]  Frank Neese,et al.  Low-energy spectrum of iron-sulfur clusters directly from many-particle quantum mechanics. , 2014, Nature chemistry.

[53]  Giovanni Li Manni,et al.  Systematic Expansion of Active Spaces beyond the CASSCF Limit: A GASSCF/SplitGAS Benchmark Study. , 2015, Journal of chemical theory and computation.

[54]  C. C. J. Roothaan,et al.  Self-Consistent Field Theory for Open Shells of Electronic Systems , 1960 .

[55]  V. Fock,et al.  Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems , 1930 .

[56]  Laura Gagliardi,et al.  The restricted active space followed by second-order perturbation theory method: theory and application to the study of CuO2 and Cu2O2 systems. , 2008, The Journal of chemical physics.

[57]  G H Loew,et al.  Role of the heme active site and protein environment in structure, spectra, and function of the cytochrome p450s. , 2000, Chemical reviews.

[58]  Qiming Sun,et al.  Stochastic Multiconfigurational Self-Consistent Field Theory. , 2015, Journal of chemical theory and computation.

[59]  A. Banerjee,et al.  Multiconfiguration wavefunctions for excited states. Selection of optimal configurations: The b 1Σ+ and d 1Σ+ states of NH , 1977 .

[60]  Ali Alavi,et al.  Breaking the carbon dimer: the challenges of multiple bond dissociation with full configuration interaction quantum Monte Carlo methods. , 2011, The Journal of chemical physics.

[61]  Rodney J. Bartlett,et al.  Similarity transformed equation-of-motion coupled-cluster study of ionized, electron attached, and excited states of free base porphin , 1997 .

[62]  B. Roos,et al.  The complete active space SCF (CASSCF) method in a Newton–Raphson formulation with application to the HNO molecule , 1981 .

[63]  K. N. Solovyov,et al.  Manifestation of the vibronic analogue of the Fermi resonance in quasi-line spectra of porphyrins: Experiment and theoretical analysis , 2007 .

[64]  Donald G Truhlar,et al.  Multiconfiguration Pair-Density Functional Theory Outperforms Kohn-Sham Density Functional Theory and Multireference Perturbation Theory for Ground-State and Excited-State Charge Transfer. , 2015, Journal of chemical theory and computation.

[65]  Ali Alavi,et al.  Linear-scaling and parallelisable algorithms for stochastic quantum chemistry , 2013, 1305.6981.

[66]  Alan Aspuru-Guzik,et al.  Quantum Monte Carlo for electronic excitations of free-base porphyrin. , 2004, The Journal of chemical physics.

[67]  Ali Alavi,et al.  Fermion Monte Carlo without fixed nodes: a game of life, death, and annihilation in Slater determinant space. , 2009, The Journal of chemical physics.

[68]  Ulf Ryde,et al.  How O2 Binds to Heme , 2004, Journal of Biological Chemistry.

[69]  G. Chan,et al.  Entangled quantum electronic wavefunctions of the Mn₄CaO₅ cluster in photosystem II. , 2013, Nature chemistry.

[70]  K. Hirao,et al.  Theoretical Study of the Q and B Bands of Free-Base, Magnesium, and Zinc Porphyrins, and Their Derivatives , 1999 .

[71]  Ali Alavi,et al.  Krylov-Projected Quantum Monte Carlo Method. , 2015, Physical review letters.

[72]  D. Dixon,et al.  Density functional theory prediction of the second-order hyperpolarizability of metalloporphines , 1995 .

[73]  M. Reiher,et al.  Density matrix renormalization group calculations on relative energies of transition metal complexes and clusters. , 2008, The Journal of chemical physics.

[74]  T. Mallouk,et al.  Metal-free organic sensitizers for use in water-splitting dye-sensitized photoelectrochemical cells , 2015, Proceedings of the National Academy of Sciences.

[75]  Christopher A. Reed,et al.  Spin-state/stereochemical relationships in iron porphyrins: implications for the hemoproteins , 1981 .

[76]  B. Roos,et al.  A Comparison of the Super-CI and the Newton-Raphson Scheme in the Complete Active Space SCF Method , 1980 .

[77]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[78]  Teizo Kitagawa,et al.  The resonance Raman spectra of intermediate-spin ferrous porphyrin , 1979 .

[79]  C. Reed,et al.  Synthesis, stereochemistry, and structure-related properties of alpha, beta, gamma, delta-tetraphenylporphinatoiron(II). , 1975, Journal of the American Chemical Society.

[80]  Garnet Kin-Lic Chan,et al.  Excited-State Geometry Optimization with the Density Matrix Renormalization Group, as Applied to Polyenes. , 2015, Journal of chemical theory and computation.

[81]  Kerstin Andersson,et al.  Second-order perturbation theory with a CASSCF reference function , 1990 .

[82]  J. Waluk,et al.  Electronic spectra and symmetry of metalloporphyrins in low-temperature rare gas and nitrogen matrices , 1997 .

[83]  Björn O. Roos,et al.  The complete active space SCF method in a fock‐matrix‐based super‐CI formulation , 2009 .

[84]  S. Grimme,et al.  DFT/MRCI calculations on the excited states of porphyrin, hydroporphyrins, tetrazaporphyrins and metalloporphyrins , 2001 .

[85]  S. Shaik,et al.  Multireference and multiconfiguration ab initio methods in heme-related systems: what have we learned so far? , 2011, The journal of physical chemistry. B.

[86]  P. Boyd,et al.  Paramagnetic anisotropy, average magnetic susceptibility, and electronic structure of intermediate-spin S = 1 (5,10,15,20-tetraphenylporphyrin)iron(II) , 1979 .

[87]  Hiroshi Nakatsuji,et al.  Ground and excited states of Mg porphin studied by the SAC/SAC-CI method , 1996 .

[88]  Carl C. Wamser,et al.  Porphyrins and phthalocyanines in solar photovoltaic cells , 2010 .

[89]  C J Umrigar,et al.  Semistochastic projector Monte Carlo method. , 2012, Physical review letters.

[90]  Sason Shaik,et al.  Theoretical perspective on the structure and mechanism of cytochrome P450 enzymes. , 2005, Chemical reviews.

[91]  Tomasz Borowski,et al.  Modeling enzymatic reactions involving transition metals. , 2006, Accounts of chemical research.

[92]  Clemens C. J. Roothaan,et al.  New Developments in Molecular Orbital Theory , 1951 .

[93]  M. Head‐Gordon,et al.  Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group , 2002 .

[94]  D. Nocera,et al.  Electocatalytic water oxidation by cobalt(III) hangman β-octafluoro corroles. , 2011, Journal of the American Chemical Society.

[95]  Juergen Hinze,et al.  MC-SCF. I. The multi-configuration self-consistent-field method , 1973 .

[96]  Alistair P. Rendell,et al.  The restricted active space self-consistent-field method, implemented with a split graph unitary group approach , 1990 .

[97]  Francesco Aquilante,et al.  SplitGAS Method for Strong Correlation and the Challenging Case of Cr2. , 2013, Journal of chemical theory and computation.