A simplified time-dependent density functional theory approach for electronic ultraviolet and circular dichroism spectra of very large molecules
Graphical abstract
Introduction
Kohn-Sham density functional theory (KS-DFT) is now the most widely used method for electronic structure calculations of larger molecules in the electronic ground state. For the calculation of excited state properties and electronic spectra of fairly large systems (about 100 atoms), time-dependent density functional theory (TD-DFT) [1], [2], [3], [4] based on a KS-DFT ground state determinant has become the most important method (see e.g. Refs. [5], [6], [7], [8], [9] for reviews). For TD-DFT calculations of ground state properties like dispersion coefficients see e.g. Refs. [10], [11].
It is generally assumed in TD-DFT that the density changes only slowly with time. For this reason, the time-dependent exchange-correlation (XC) kernel is replaced by a time-independent one which is then evaluated using the time-dependent density. This so-called adiabatic approximation [1], [2] allows one to make use of ground state XC potentials in TD-DFT. The states of common interest are often valence states far below the first ionization potential for which this approximation works fairly well [3].
The accuracy of TD-DFT for vertical excitation energies is roughly comparable to that of KS-DFT for relative ground state energies (about 0.2–0.3 eV for TD-DFT, 2–5 kcal/mol for KS-DFT) [12], [13]. There are some well-known deficiencies of TD-DFT such as the description of excitations with double excitation or multiplet character which are problematic for any single-reference method. Density functionals of the general gradient approximation (GGA) type, have an XC potential of an incorrect asymptotic form [7] and suffer also from the self-interaction error (SIE) [14], [15], [16], [17] and the related integer discontinuity problem [18], [17]. General implications are underestimated ionization potentials, overestimated electron affinities and too small energy gaps between Kohn-Sham orbitals. Due to these deficiencies TD-DFT/GGA severely underestimates charge-transfer (CT) and Rydberg excitations [7], [19], [20], [18], [21]. This artifically introduces many states with low oscillator strength (‘ghost states’) to the low energetic part of the spectrum. Even though they are often not observed in a simulated ultraviolet-visible (UV-Vis) spectrum, they may cause artificial mixing of configurations and ‘contaminate’ the bright states which can then corrupt spectra considerably.
Since Hartree-Fock (HF) does not suffer from SIE and exhibits the correct asymptotic potential, admixing certain amounts of non-local Fock exchange to the GGA exchange, as done in global hybrids (e.g. B3LYP [22], [23], [24]), partially alleviates the SIE as well as the CT problem in TD-DFT. Such functionals are more reliable in an excited state treatment than pure GGAs on the one hand and HF based single-excitation methods on the other. The advantage of (hybrid) TD-DFT over the latter are the implicitly ‘correlated’ Kohn-Sham orbitals. Since semi-empirical methods are typically less reliable and wave-function based methods which include double and higher excitations are too costly to be applied to large systems, hybrid TD-DFT has become one of the most widely used methods to describe excited states systems up to about 100 atoms.
The amount of non-local Fock exchange for typical hybrid functionals lies in the range of 10–25%, but can be as high as 50% like in the BHLYP functional [22]. In a systematic study, it has been found that on average admixing 40% of non-local Fock exchange yields the best excitation energies of fairly large organic molecules [25], although the correct treatment of some larger systems required higher amounts of Fock exchange up to 50% or more. A commonly applied class of functionals in TD-DFT calculations nowadays are the range-separated-hybrid (RSH) functionals that, starting with no or low amounts of Fock exchange, asymptotically employ 100% non-local Fock exchange (65% in the case of CAM-B3LYP) [26], [27], [28], [29]. With these functionals, the correct asymptotic behaviour is achieved. Nevertheless, we will employ exclusively global hybrids in this work and make use of the BHLYP functional in large systems where the description of CT states may become problematic. The extension of the here proposed method to RSH functionals is straightforward (see below) and will be discussed elsewhere.
Even though TD-DFT can deal with systems beyond the scope of traditional wave function based methods, the theoretical treatment of an entire UV-Vis electronic spectrum in a typical excitation energy range from 2 to 7 eV for systems with several hundreds up to about 1000 atoms remains a challenge. Recently, a simplified Tamm-Dancoff approach to time-dependent density functional theory (sTDA) has been proposed which allows routine computations of UV-Vis or circular dichroism (CD) spectra of such large systems [30]. The drastic simplifications are on the one hand, the evaluation of the two-electron integrals as short-range damped Coulomb interactions between (transition) charge density monopoles and a massive truncation of the single excitation expansion space on the other. Solving the Tamm-Dancoff approximated problem (TDA) [31], [32] instead of the TD-DFT equation requires the solution of only one eigenvalue problem and along with the simplifications mentioned above, this makes the sTDA approach extremely fast even for large molecules.
While the TDA typically gives quite similar excitation energies to TD-DFT, it suffers from the fact that it is not gauge invariant and oscillator and rotatory strengths obtained from TDA do not satisfy the respective sum rules [4], [7], [33]. These shortcomings can become particularly problematic if one is interested in calculating rotator strengths for CD spectra. TD-DFT, on the other hand, does not suffer from these shortcomings. Therefore, a simplified time-dependent density functional theory approach (sTD-DFT) is presented here which makes use of the same simplifications as made in sTDA while the full TD-DFT problem is solved.
This paper is structured in the following way: After a brief outline of the basic theory followed by a short summary of the sTDA method, we will recapitulate the dipole length and the dipole velocity formalisms for transition moments. Then the sTD-DFT is discussed. Finally, we will compare the performance of sTD-DFT to sTDA and conventional TD-DFT for the computation of excitation energies as well as UV and CD spectra for various systems.
Section snippets
TD-DFT and TDA
The full TD-DFT response problem is given by the following non-Hermitian eigenvalue problem [1], [7].where A and B are the so-called orbital rotation Hessian matrices with eigenfunctions X and Y and is a vector with the dimension of the number of roots (nroots) that contains the respective eigenvalues. In wave function theory, this equation corresponds to time-dependent Hartree-Fock (TD-HF, also called random-phase approximation, RPA). For a global hybrid density functional in
Technical details of the calculations
We used the TURBOMOLE suite of programs [53], [54] (version 6.5) to perform all ground state DFT calculations. The sTD-DFT was implemented into our group-intern sTDA code which is a stand-alone program that is compatible with TURBOMOLE. All of the sTDA or sTD-DFT excited state calculations presented here were carried out with this version of the program. We employ standard integration grids (m4 if not noted otherwise) and typical SCF convergence criteria (). With the exception of Section
Rotatory strengths from conventional and simplified TDA and TD-DFT in the biphenyl molecule
In this section, the inherent deficiency of TDA in obtaining reliable rotatory strengths as well as oscillator strengths will be demonstrated using the biphenyl molecule as a test case. As a measure of internal consistency of each method, we consider the difference of the rotatory strengths obtained from the length and velocity formalism (Eqs. (19), (20)). We will thus use the modulus of the difference , which is zero for exact wave functions.
For biphenyl as a test case (Fig. 1),
Conclusions
A simplified approach to TD-DFT has been presented which is based on the same simplifications as in the recently published sTDA method [30]. Without refitting of parameters, reliable UV and CD spectra can now routinely be obtained for systems with up to 1000 atoms. The compatibility of sTDA parameters with sTD-DFT was shown by examining VEEs obtained from sTD-DFT. This makes sTD-DFT a valuable and (parameter) consistent extension to sTDA which we recommend to be used in order to obtain higher
Acknowledgments
This work was supported by the Fonds der Chemischen Industrie and the DFG in the framework of the SFB 813 (‘Chemistry at Spin-Centers’). C. B. thanks P. Shushkov for helpful discussions during the implementation of the sTD-DFT code which is implemented in a stand-alone program and which can be obtained from our website [73].
References (73)
- et al.
Treatment of electronic excitations within the adiabatic approximation of time dependent density functional theory
Chem. Phys. Lett
(1996) - et al.
Calculation of excitation energies of organic chromophores: a critical evaluation
J. Mol. Struct. (Theochem)
(2002) - et al.
Combining long–range configuration interaction with short–range density functionals
Chem. Phys. Lett.
(1997) - et al.
A new hybrid exchange–correlation functional using the Coulomb–attenuating method (CAM-B3LYP)
Chem. Phys. Lett
(2004) - et al.
Time–dependent density functional theory within the Tamm–Dancoff approximation
Chem. Phys. Lett.
(1999) Density functional calculations with configuration interaction for the excited states of molecules
Chem. Phys. Lett.
(1996)- et al.
Electronic structure calculations on workstation computers: the program system turbomole
Chem. Phys. Lett.
(1989) - et al.
Integral approximations for LCAO–SCF calculations
Chem. Phys. Lett.
(1993) - et al.
Enantiomerically pure [M6L12] or [M12L24] polyhedra from flexible bis(pyridine) ligands
Angew. Chem. Int. Ed.
(2014)et al.Enantiomerenreine [M6L12]- oder [M12L24]-Polyeder aus flexiblen Bis(pyridin)-Liganden
Angew. Chem.
(2014) Time-dependent density functional response theory for molecules
Density functional theory of time–dependent phenomena
Top. Curr. Chem.
On the density matrix based approach to time–dependent density functional response theory
J. Chem. Phys.
Spectroscopy: computational methods
Calculation of the electronic spectra of large molecules
Single-reference ab initio methods for the calculation of excited states of large molecules
Chem. Rev.
The art of choosing the right quantum chemical excited-state method for large molecular systems
A density functional theory study of frequency-dependent polarizabilities and Van der Waals dispersion coefficients for polyatomic molecules
J. Chem. Phys.
Density functional results for isotropic and anisotropic multipole polarizabilities and and Van der Waals dispersion coefficients for molecules
J. Chem. Phys.
A thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions
Phys. Chem. Chem. Phys.
Incorrect dissociation behavior of radical ions in density functional calculations
J. Phys. Chem. A
A challenge for density functionals: self-interaction error increases for systems with a noninteger number of electrons
J. Chem. Phys.
Comparison of the accurate Kohn–Sham solution with the Generalized Gradient Approximations (GGAs) for the SN2 Reaction F− + CH3F → FCH3 + F−: a qualitative rule to predict success or failure of GGAs
J. Phys. Chem. A
Challenges for density functional theory
Chem. Rev.
Relationship between long-range charge-transfer excitation energy error and integer discontinuity in Kohn–Sham theory
J. Chem. Phys.
Does density functional theory contribute to the understanding of excited states of unsaturated organic compounds?
Mol. Phys.
Long-range charge-transfer excited states in time-dependent density functional theory require non-local exchange
J. Chem. Phys.
Failure of time-dependent density functional theory for long-range charge-transfer excited states: the zincbacteriochlorin-bacteriochlorin and bacteriochlorophyll-spheroidene complexes
J. Am. Chem. Soc.
A new mixing of Hartree–Fock and local density-functional theories
J. Chem. Phys.
Density-functional thermochemistry. III. The role of exact exchange
J. Chem. Phys.
Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields
J. Phys. Chem.
The vibronic structure of electronic absorption spectra of large molecules: a time-dependent density functional study on the influence of “Exact” Hartree–Fock exchange
J. Phys. Chem. A
Coulomb-attenuated exchange energy density functionals
Mol. Phys.
A long-range correction scheme for generalized-gradient-approximation exchange functionals
J. Chem. Phys.
A simplified Tamm–Dancoff density functional approach for the electronic excitation spectra of very large molecules
J. Chem. Phys.
Quantum Theory of Many-Particle Systems
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