Adiabatic approximation of time-dependent density matrix functional response theory.

Time-dependent density matrix functional theory can be formulated in terms of coupled-perturbed response equations, in which a coupling matrix K(omega) features, analogous to the well-known time-dependent density functional theory (TDDFT) case. An adiabatic approximation is needed to solve these equations, but the adiabatic approximation is much more critical since there is not a good "zero order" as in TDDFT, in which the virtual-occupied Kohn-Sham orbital energy differences serve this purpose. We discuss a simple approximation proposed earlier which uses only results from static calculations, called the static approximation (SA), and show that it is deficient, since it leads to zero response of the natural orbital occupation numbers. This leads to wrong behavior in the omega-->0 limit. An improved adiabatic approximation (AA) is formulated. The two-electron system affords a derivation of exact coupled-perturbed equations for the density matrix response, permitting analytical comparison of the adiabatic approximation with the exact equations. For the two-electron system also, the exact density matrix functional (2-matrix in terms of 1-matrix) is known, enabling testing of the static and adiabatic approximations unobscured by approximations in the functional. The two-electron HeH(+) molecule shows that at the equilibrium distance, SA consistently underestimates the frequency-dependent polarizability alpha(omega), the adiabatic TDDFT overestimates alpha(omega), while AA improves upon SA and, indeed, AA produces the correct alpha(0). For stretched HeH(+), adiabatic density matrix functional theory corrects the too low first excitation energy and overpolarization of adiabatic TDDFT methods and exhibits excellent agreement with high-quality CCSD ("exact") results over a large omega range.

[1]  R. Parr,et al.  Elementary properties of an energy functional of the first‐order reduced density matrix , 1978 .

[2]  M. Grüning,et al.  On the required shape corrections to the local density and generalized gradient approximations to the Kohn-Sham potentials for molecular response calculations of (hyper)polarizabilities and excitation energies , 2002 .

[3]  Kieron Burke,et al.  Time-dependent density functional theory: past, present, and future. , 2005, The Journal of chemical physics.

[4]  E. Baerends,et al.  Excitation energies of dissociating H2: A problematic case for the adiabatic approximation of time-dependent density functional theory , 2000 .

[5]  G. Zumbach,et al.  Density‐matrix functional theory for the N‐particle ground state , 1985 .

[6]  D. Chong.  Augmenting basis set for time-dependent density functional theory calculation of excitation energies: Slater-type orbitals for hydrogen to krypton , 2005 .

[7]  B. A. Hess,et al.  The structure of the second-order reduced density matrix in density matrix functional theory and its construction from formal criteria. , 2004, The Journal of chemical physics.

[8]  D. Chong.  Recent Advances in Density Functional Methods Part III , 2002 .

[9]  M. Buijse,et al.  Electron correlation fermi and coulomb holes dynamical and nondynamical correlation , 1991 .

[10]  Evert Jan Baerends,et al.  An approximate exchange-correlation hole density as a functional of the natural orbitals , 2002 .

[11]  E. Baerends,et al.  Coupled-perturbed density-matrix functional theory equations. Application to static polarizabilities. , 2006, The Journal of chemical physics.

[12]  H. Koch,et al.  Integral-direct coupled cluster calculations of frequency-dependent polarizabilities, transition probabilities and excited-state properties , 1998 .

[13]  Andreas Dreuw,et al.  Single-reference ab initio methods for the calculation of excited states of large molecules. , 2005, Chemical reviews.

[14]  Evert Jan Baerends,et al.  Density-functional-theory response-property calculations with accurate exchange-correlation potentials , 1998 .

[15]  J. G. Snijders,et al.  Improved density functional theory results for frequency‐dependent polarizabilities, by the use of an exchange‐correlation potential with correct asymptotic behavior , 1996 .

[16]  M. Head‐Gordon,et al.  Long-range charge-transfer excited states in time-dependent density functional theory require non-local exchange , 2003 .

[17]  M. E. Casida.  Time-Dependent Density Functional Response Theory for Molecules , 1995 .

[18]  Ian J. Bush,et al.  The GAMESS-UK electronic structure package: algorithms, developments and applications , 2005 .

[19]  Harrison Shull,et al.  NATURAL ORBITALS IN THE QUANTUM THEORY OF TWO-ELECTRON SYSTEMS , 1956 .

[20]  T. H. Dunning.  Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen , 1989 .

[21]  T. Gilbert.  Hohenberg--Kohn theorem for nonlocal external potentials , 1975 .

[22]  O. Heinonen,et al.  Many-Particle Theory, , 1991 .

[23]  F. Gadéa,et al.  Charge-transfer correction for improved time-dependent local density approximation excited-state potential energy curves: Analysis within the two-level model with illustration for H2 and LiH , 2000 .

[24]  S. Valone,et al.  Consequences of extending 1‐matrix energy functionals from pure–state representable to all ensemble representable 1 matrices , 1980 .

[25]  Katarzyna Pernal,et al.  Effective potential for natural spin orbitals. , 2005, Physical review letters.

[26]  Á. Rubio,et al.  Time-dependent density-functional theory. , 2009, Physical chemistry chemical physics : PCCP.

[27]  David E. Woon,et al.  Gaussian basis sets for use in correlated molecular calculations. IV. Calculation of static electrical response properties , 1994 .

[28]  Evert Jan Baerends,et al.  An improved density matrix functional by physically motivated repulsive corrections. , 2005, The Journal of chemical physics.

[29]  K. Tamura,et al.  Metabolic engineering of plant alkaloid biosynthesis. Proc Natl Acad Sci U S A , 2001 .

[30]  Evert Jan Baerends,et al.  Molecular calculations of excitation energies and (hyper)polarizabilities with a statistical average of orbital model exchange-correlation potentials , 2000 .

[31]  E. Baerends,et al.  Time-dependent density-matrix-functional theory , 2007 .

[32]  M. Piris,et al.  Assessment of a new approach for the two-electron cumulant in natural-orbital-functional theory. , 2005, The Journal of chemical physics.

[33]  G. Herzberg,et al.  Constants of diatomic molecules , 1979 .

[34]  J. Cioslowski,et al.  Ionization potentials from the extended Koopmans’ theorem applied to density matrix functional theory , 2005 .

[35]  Evert Jan Baerends,et al.  Excitation energies of metal complexes with time-dependent density functional theory , 2004 .

[36]  M. Levy.  Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. , 1979, Proceedings of the National Academy of Sciences of the United States of America.

[37]  Evert Jan Baerends,et al.  Asymptotic correction of the exchange-correlation kernel of time-dependent density functional theory for long-range charge-transfer excitations. , 2004, The Journal of chemical physics.

[38]  S. Goedecker,et al.  Natural Orbital Functional for the Many-Electron Problem , 1998 .