Embedding a quantum mechanical system in orbital-free environment:
Overview
In modelling complex polyatomic systems, the most common strategy relies on approximate solutions to the Schrodinger equation. Various methods have been developed for this purpose. One-electron functions (orbitals) are used in such methods for various purpose. For instance, they are used to approximate the exact wavefunction in Hartree-Fock theory whereas they are used to construct the wavefunction of the fictitious system of non-interacting electrons in the Kohn-Sham formalism. The number of such orbitals increases at least linearly (the case of Kohn-Sham formalism) with the size of the system. As a result, the computational costs any practical computational method in which the orbitals are handled (derived, orthogonalized, transformed, stored, etc.) increases faster than linear with the size of the system. This leaves mesoscopic and macroscopic systems outside of the domain of applicability of such quantum mechanics based methods which use the orbitals.
Our long-standing interests concern an universal first-principles based formalism in which the quantum mechanical level of description is retained for the whole system of any size but the orbitals are used only for a selected subsystem (a solvated molecule for instance). In this formalism, the orbitals used to construct the electron density of the selected subsystem are obtained from Kohn-Sham-like one-electron equations [Eqs. 20-21, in Wesolowski and Warshel, J. Phys. Chem. 97 (1993) 8050]. The fact that the studied subsystem is embedded in a microscopic environment is represented in this formalism by means of a special term in the effective potential - effective embedding potential. This embedding potential is system-independent because it uses the universal functionals of electron density. It is expressed as a function of two variables ρA and ρB which are the electron densities of the embedded system and its microscopic environment, respectively. It is, therefore, orbital-free.
The explicit analytic form of the orbital-free embedding effective potential is not known but it can be expressed using approximations to the non-additive kinetic energy functional (the difference between the kinetic energy Ts[ρ] of the total electron densities and that of the separated electron densities of the two subsystems) and the exchange-correlation functional defined in the Kohn-Sham formalism. The terms known as 'coupling term', 'coupling potential', etc. in hybrid QM/MM methods can be seen thus as empirical parametrisations of the effective embedding potential given in Eqs. 20-21 in Wesolowski and Warshel, J. Phys. Chem. 97 (1993) 8050].
The publications concerning the key methodological and computer implementation steps relevant to the orbital-free embedding formalism are listed below.
Embedding a quantum mechanical system in orbital-free environment:
Key Publications
1) One-electron equations for embedded orbitals are introduced in this work (Eqs. 20-21):
Frozen Density Functional Approach for ab-initio Calculations of Solvated Molecules
T.A. Wesolowski and A. Warshel J. Phys. Chem. 97 (1993) 8050
2) Incorporating one-electron equations for embedded orbitals into the linear-response time-dependent DFT framework
Generalization of the Kohn-Sham Equations with Constrained Electron Density (KSCED) Formalism and its Time-Dependent Response Theory Formulation.
M. Casida and T.A. Wesolowski Intl. J. Quant. Chem. 96 (2004) 577-588
3) Derivation of the universal orbital-free embedding potential for wavefunction based methods:
Embedding a multi-determinantal wavefunction in orbital-free environment.T.A. Wesolowski, Phys. Rev.A. 77 (2008) 012504.
4) A general review:
One-electron Equations for Embedded Electron Density: Challenge for Theory and Practical Payoffs in Multi-Level Modeling of Complex Polyatomic SystemsT.A. Wesolowski, Computational Chemistry: Reviews of Current Trends - Vol. 10 World Scientific, 2006, pp. 1-82.
Embedding a quantum mechanical system in orbital-free environment:
Specific issues
Testing approximations to the kinetic-energy-functional dependent part of the effective embedding potential given in [Eq.21 Wesolowski,Warshel, J. Phys. Chem., 97 (1993) 8050]:
Kohn-Sham equations with constrained electron density: The effect of various kinetic energy functional parametrizations on the ground-state molecular properties.T.A. Wesolowski and J. Weber, Intl. J. Quant. Chem. 61 (1997)303.
Accuracy of Approximate Kinetic Energy Functionals in the Model of Kohn-Sham Equations with Constrained Electron Density: the FH...NCH complex as a Test Case
T.A. Wesolowski, H. Chermette, and J. Weber, J. Chem. Phys. 105 (1996)9182.
Density Functional Theory with approximate kinetic energy functionals applied to hydrogen bonds
T.A. Wesolowski, J. Chem. Phys. 106 (1997)8516.
Exact properties of the bi-functional of the nonadditive kinetic energy:
Exact inequality involving the kinetic energy functional Ts[ρ] and pairs of electron densities.
T.A. Wesolowski Journal of Physics A: Mathematical and General: 36 (2003) 10607
Approximating the kinetic energy functional Ts[rho]: lessons from four-electron systems.
T.A. Wesolowski, Mol. Phys. 103 (2005) 1165-1167. (Handy special issue)
Orbital-free effective embedding potential at nuclear cuspsJ.-M. Garcia Lastra, Jakub W. Kaminski, and Tomasz A. Wesolowski, J. Chem. Phys. 129 (2008) 074107.
Orbital-free embedding effective potential in analytically solvable casesAndreas Savin, Tomasz A. Wesolowski Progress in Theoretical Chemistry and Physics, 19 (2009) 327-339.
Full minimization of the total energy bi-functional E[ρA,ρB] ("freeze-and-thaw" cycle, relation to the model of Kim and Gordon, performance of generalized gradient approximation in Kohn-Sham calculations and in the minimization of E[ρA,ρB]):
Kohn-Sham equations with constrained electron density: an iterative evaluation of the ground-state electron density of interacting molecules
T.A. Wesolowski and J. Weber, Chem. Phys. Lett., 248 (1996)71
Intermolecular interaction energies from the total energy bi-functional. A case study of carbazole complexes.
T.A. Wesolowski, P.-Y. Morgantini, and J. Weber J. Chem. Phys. 116 (2002) 6411
Gradient-free and gradient-dependent approximations in the total energy bi-functional for weakly overlapping electron densities.
T.A. Wesolowski and F. Tran, J. Chem. Phys. 118 (2003) 2072
Electronic excitations of embedded molecules obtained from the formalism merging orbital-free embedding and linear-response time-dependent DFT [Casida and Wesolowski, Intl. J. Quant. Chem. , 96 (2004) 577-588]:
Hydrogen-bonding induced shifts of the excitation energies in nucleic acid bases: an interplay between electrostatic- and electron density overlap effects.T.A. Wesolowski, J. Am. Chem. Soc. 126 (2004) 11444-11445.
Embedding vs supermolecular strategies in evaluating the hydrogen-bonding-induced shifts of excitation energiesGeorgios Fradelos, Jesse J. Lutz, Tomasz A. Wesolowski, Piotr Piecuch, and Marta Wloch Journal of Chemical Theory and Computations, 7 (2011) 1647-1666.
The merits of the frozen-density embedding scheme to model solvatochromic shifts.
J. Neugebauer, M.J. Louwerse, E.J. Baerends, T.A. Wesolowski, J. Chem. Phys. 122 (2005) 094115
Non-uniform continuum solvent model for solvatochromism
Modeling solvatochromic shifts using the orbital-free embedding potential at statistically-mechanically averaged solvent densityJakub W. Kaminski, Sergey Gusarov, Andriy Kovalenko, Tomasz A. Wesolowski Journal of Physical Chemistry A , 114 (2010) 6082.
Multi-scale modelling of solvatochromic shifts from frozen-density embedding the ory with non-uniform continuum model of the solvent: the coumarin 153Xiuwen Zhou, Jakub W. Kaminski, Tomasz A. Wesolowski Phys. Chem. Chem. Phys. , 13 (2011) 10565-10576 (special issue on Multilevel Modelling).
The use of the orbital-free embedding effective potential given in [Eq.21 Wesolowski,Warshel, J. Phys. Chem., 97 (1993) 8050] in the QM/MD simulations of solvated molecules to couple the quantum subsystem (QM) with the subsystem described using classical molecular mechanics (MD):
Free Energy Perturbation Calculations of Solvation Free Energy Using Frozen Density Functional Approach.
T.A. Wesolowski and A. Warshel, J. Phys. Chem. 98 (1994) 5183
Implementation of analytic gradients (geometry optimisation and harmonic frequency calculations):
Properties of CO adsorbed in ZSM5 Zeolite. Density Functional Theory Study Using the Embedding Scheme Based on Electron Density Partitioning.
T.A. Wesolowski, A. Goursot, and J. Weber, J. Chem. Phys. 115 (2001) 4791
Basis set dependence of the results of variational calculations using the total energy bi-functional (geometry optimisation and harmonic frequency calculations):
Basis set effect on the results of the minimization of the total energy bifunctional E[ρA,ρB].
M. Dulak, T.A. Wesolowski Intern. J. Quant. Chem. 101 (2005) 543-549
Spin-polarized version of the formalism (hyperfine structure):
Application of the DFT based embedding scheme using explicit functional of the kinetic energy to determine the spin-density of Mg+ embedded in Ne and Ar matrices
T.A. Wesolowski, Chem. Phys. Lett. 311 (1999) 87