Analytical derivative techniques for TDDFT excited-state properties: Theory and application

We review our recent work on the methodology development of the excited-state properties for the molecules in vacuum and liquid solution.The general algorithms of analytical energy derivatives for the specific properties such as the first and second geometrical derivatives and IR/Raman intensities are demonstrated in the framework of the time-dependent density functional theory(TDDFT).The performance of the analytical approaches on the calculation of excited-state energy Hessian has also been shown.It is found that the analytical approaches are superior to the finite-difference method on the computational accuracy and efficiency.The computational cost for a TDDFT excited-state Hessian calculation is only 2–3 times as that for the DFT ground-state Hessian calculation.With the low computational complexity of the developed analytical approaches,it becomes feasible to realize the large-scale numerical calculations on the excited-state vibrational frequencies,vibrational spectroscopies and the electronic-structure parameters which enter the spectrum calculations of electronic absorption and emission,and resonance Raman spectroscopies for medium-to large-sized systems.     

Analytical approach for the excited-state Hessian in time-dependent density functional theory: formalism, implementation, and performance

The paper presents the formalism, implementation, and performance of the analytical approach for the excited-state Hessian in the time-dependent density functional theory (TDDFT) that extends our previous work [J. Liu and W. Z. Liang, J. Chem. Phys. 135, 014113 (2011)] on the analytical Hessian in TDDFT within Tamm-Dancoff approximation (TDA) to full TDDFT. In contrast to TDA-TDDFT, an appreciable advantage of full TDDFT is that it maintains the oscillator strength sum rule, and therefore yields more precise results for the oscillator strength and other related physical quantities. For the excited-state harmonic vibrational frequency calculation, however, full TDDFT does not seem to be advantageous since the numerical tests demonstrate that the accuracy of TDDFT with and without TDA are comparable to each other. As a common practice, the computed harmonic vibrational frequencies are scaled by a suitable scale factor to yield good agreement with the experimental fundamental frequencies. Here we apply both the optimized ground-state and excited-state scale factors to scale the calculated excited-state harmonic frequencies and find that the scaling decreases the root-mean-square errors. The optimized scale factors derived from the excited-state calculations are slightly smaller than those from the ground-state calculations.     

Resonance Raman spectra of uracil based on Kramers-Kronig relations using time-dependent density functional calculations and multireference perturbation theory

The use of time-dependent density functional calculations for the optimization of excited-state structures and the subsequent calculation of resonance Raman intensities within the transform-theory framework is compared to calculations of Hartree-Fock/configuration interaction singles-type (CIS). The transform theory of resonance Raman scattering is based on Kramers-Kronig relations between polarizability tensor components and the optical absorption. Stationary points for the two lowest excited singlet states of uracil are optimized and characterized by means of numerical differentiation of analytical excited-state gradients. It is shown that the effect of electron correlation leads to substantial modifications of the relative intensities. Calculations of vibrational frequencies for ground and excited states are carried out, which show that the neglect of Duschinsky mixing and the assumption of equal wave numbers for ground and excited state are not in all cases good approximations. We also compare the transform-theory resonance Raman intensities with those obtained within a simple approximation from excited-state gradients at the ground-state equilibrium position, and find that they are in qualitative agreement in the case of CIS, but show some important differences in calculations based on density functional theory. Since the results from CIS calculations are in better agreement with experiment, we also present approximate resonance Raman spectra obtained using excited-state gradients from multireference perturbation theory calculations, which confirm the CIS gradients.     

Analytical energy derivatives and normal modes in force fields employing lone-pair pseudoatoms

Formulas are derived for analytical first and second energy derivatives with respect to nuclear coordinates in molecular mechanics force fields employing lone-pair pseudoatoms. These derivatives may further be used for the calculation of normal modes and vibrational frequencies while properly accounting for the presence of pseudoatoms. The equations are applied using the MM 2 force field to calculate the vibrational spectrum of methanol to illustrate the applicability of the method. The results are compared to both experiment and a numerical approximation in which small masses are assigned to the lone-pair pseudoatoms.     

Structural and spectral properties of 4-bromo-1-naphthyl chalcones: a quantum chemical study

The structural and optical properties of 4-bromo-1-naphthyl chalcones (BNC) have been studied by using quantum chemical methods. The density functional theory (DFT) and the singly excited configuration interaction (CIS) methods were employed to optimize the ground and excited state geometries of unsubstituted and substituted BNC with different electron withdrawing and donating groups in both gas and solvent phases. Based on the ground and excited state geometries, the absorption and emission spectra of BNC molecules were calculated using the time-dependent density functional theory (TDDFT) method. The solvent phase calculations were performed using the polarizable continuum model (PCM). The geometrical parameters, vibrational frequencies, and relative stability of cis- and trans-isomers of unsubstituted and substituted BNC molecules have been studied. The results from the TDDFT calculations reveal that the substitution of electron withdrawing and electron donating groups affects the absorption and emission spectra of BNC.     

Analytical time-dependent density functional derivative methods within the RI-J approximation, an approach to excited states of large molecules

Time-dependent density functional theory (TDDFT) is now well established as an efficient method for molecular excited state treatments. In this work, we introduce the resolution of the identity approximation for the Coulomb energy (RI-J) to excited state gradient calculations. In combination with nonhybrid functionals, the RI-J approximation leads to speed ups in total timings of an order of magnitude compared to the conventional method; this is demonstrated for oligothiophenes with up to 40 monomeric units and adamantane clusters. We assess the accuracy of the computed adiabatic excitation energies, excited state structures, and vibrational frequencies on a set of 36 excited states. The error introduced by the RI-J approximation is found to be negligible compared to deficiencies of standard basis sets and functionals. Auxiliary basis sets optimized for ground states are suitable for excited state calculations with small modifications. In conclusion, the RI-J approximation significantly extends the scope of applications of analytical TDDFT derivative methods in photophysics and photochemistry.     

Ultrafast infrared spectroscopy of riboflavin: dynamics, electronic structure, and vibrational mode analysis

Femtosecond time-resolved infrared spectroscopy was used to study the vibrational response of riboflavin in DMSO to photoexcitation at 387 nm. Vibrational cooling in the excited electronic state is observed and characterized by a time constant of 4.0 +/- 0.1 ps. Its characteristic pattern of negative and positive IR difference signals allows the identification and determination of excited-state vibrational frequencies of riboflavin in the spectral region between 1100 and 1740 cm (-1). Density functional theory (B3LYP), Hartree-Fock (HF) and configuration interaction singles (CIS) methods were employed to calculate the vibrational spectra of the electronic ground state and the first singlet excited pipi* state as well as respective electronic energies, structural parameters, electronic dipole moments and intrinsic force constants. The harmonic frequencies of the S 1 excited state calculated by the CIS method are in satisfactory agreement with the observed band positions. There is a clear correspondence between computed ground- and excited-state vibrations. Major changes upon photoexcitation include the loss of the double bond between the C4a and N5 atoms, reflected in a downshift of related vibrations in the spectral region from 1450 to 1720 cm (-1). Furthermore, the vibrational analysis reveals intra- and intermolecular hydrogen bonding of the riboflavin chromophore.     

Time-dependent density functional theory study of Fe2(CO)9 low-lying electronic excited states

The valence electronic excited states of Fe2(CO)9 have been studied using the time-dependent density functional theory (TDDFT). Both tribridged D3h and monobridged C2v structures have been considered, and the structure of selected low-lying singlet and triplet excited states have been optimized on the basis of the TDDFT analytical gradient. Optimized excited-state geometries are used to obtain an insight into certain aspects of the Fe2(CO)9 photochemistry. The Fe2(CO)9 (D3h) first triplet and second singlet excited states are unbound with respect to dibridged Fe2(CO)8 + CO, and the first two monobridged Fe2(CO)9 (C2v) singlet states are unbound with respect to the Fe(CO)5 + Fe(CO)4 dissociation. These results are discussed in light of the experimental data available.     

Ultrafast hydrogen bond strengthening of the photoexcited fluorenone in alcohols for facilitating the fluorescence quenching

The time-dependent density functional theory (TDDFT) method was performed to investigate the excited-state hydrogen-bonding dynamics of fluorenone (FN) in hydrogen donating methanol (MeOH) solvent. The infrared spectra of the hydrogen-bonded FN-MeOH complex in both the ground state and the electronically excited states are calculated using the TDDFT method, since the ultrafast hydrogen-bonding dynamics can be investigated by monitoring the vibrational absorption spectra of some hydrogen-bonded groups in different electronic states. We demonstrated that the intermolecular hydrogen bond C=O...H-O between fluorenone and methanol molecules is significantly strengthened in the electronically excited-state upon photoexcitation of the hydrogen-bonded FM-MeOH complex. The hydrogen bond strengthening in electronically excited states can be used to explain well all the spectral features of fluorenone chromophore in alcoholic solvents. Furthermore, the radiationless deactivation via internal conversion (IC) can be facilitated by the hydrogen bond strengthening in the excited state. At the same time, quantum yields of the excited-state deactivation via fluorescence are correspondingly decreased. Therefore, the total fluorescence of fluorenone in polar protic solvents can be drastically quenched by hydrogen bonding.     

Coupled-cluster studies of the lowest excited states of the 11-cis-retinal chromophore

The first few excited states of the 11-cis-retinal (PSB11) chromophore have been studied at the coupled-cluster approximative singles and doubles (CC2) level using triple-zeta quality basis sets augmented with double sets of polarisation functions. The two lowest vertical excitation energies of 2.14 and 3.21 eV are in good agreement with recently reported experimental values of 2.03 and 3.18 eV obtained in molecular beam measurements. Calculations at the time-dependent density functional theory (TDDFT) level using the B3LYP hybrid functional yield vertical excitation energies of 2.34 and 3.10 eV for the two lowest states. Zero-point vibrational energy (ZPVE) corrections of -0.09 and -0.17 eV were deduced from the harmonic vibrational frequencies for the ground and excited states calculated at the density functional theory (DFT) and TDDFT level, respectively, using the B3LYP hybrid functional.     

Rotational symmetry of the molecular potential energy in the Cartesian coordinates

We consider the molecular Born-Oppenheimer potential energy as a function of atomic Cartesian coordinates and discuss the non-stationary Hessian properties arising due to rotational symmetry. A connection with the extended Hessian theory is included. New applications of Cartesian representation for examining and correcting raw numerical Hessian data and a simple formulation of harmonic vibrational analysis of partially optimized systems are proposed. Exemplary calculations for the porphyrin molecule with an internal proton transfer are reported. We also develop the normal transformation method to incorporate the rotational symmetry into the approximate analytical potentials, which are parametrized in the Cartesian coordinates. The transformation converts the coordinates from the space fixed frame to the frame which translates and rotates with the molecule and is determined by the Eckart conditions. New simple analytical formulas for the first and second derivatives of the transformed potential are derived. This fast method can be used to calculate the potential and its derivatives in the simulations of chemical reaction dynamics in the space fixed Cartesian frame without the need to constrain the molecular rotation or to define the local non-redundant internal coordinates.     

A rotationally invariant procedure for density functional calculations

A simple method which is rigorously invariant under molecular rotations is presented for evaluation of the density functional exchange—correlation energy by numerical quadrature. The corresponding expressions for the first and second derivatives of the energy with respect to nuclear displacement are presented. In particular, such a scheme is necessary to remove the difficulties previously encountered in calculating Kohn—Sham harmonic vibrational frequencies for low-lying modes.     

香豆素衍生物的荧光发射能计算及XC泛函的合理选择

采用含时密度泛函理论(TDDFT)与单激发组态相互作用(CIS)处理相结合的计算方案对香豆素系列15种已知荧光化合物的发射能进行了系统考察. 结果表明, 发射能与吸收能一样, 其计算值主要取决于交换-相关(XC)泛函的选择. 只要泛函选用得当, 在使用较小基组的TDDFT/6-31G(d)//CIS/3-21G(d)理论水平上即可使绝大部分化合物的实验发射能在精度达0.16 eV以内得以重现. 与吸收能计算不同的是, 无法选用单一的一种泛函来对全系列化合物的发射能作出满意的理论预测. 激发态无明显电荷转移的、7位上有氨(或胺)基取代或有氮原子相连的化合物, 其适用泛函为不含Hartree-Fock(HF)交换能的纯泛函OLYP和BLYP. 而激发态发生较大程度电荷转移的、3 位上有共轭取代基的衍生物, 其适用泛函则为含20%的HF交换成分的混合泛函B3LYP. 因此, 发射能计算中的XC泛函选择, 应同时考虑取代基团效应以及激发态的电子结构特征. 其中, 发射能计算值受XC泛函中HF交换能比例的影响十分敏感. 文中还对激发能计算中的溶剂效应校正方案和激发态几何优化精度的影响进行了讨论.     

Proper Choice of XC Functionals and Calculations of Fluorescence-Emitting Energies for Coumarin Derivatives

A scheme of time-dependent density functional theory (TDDFT) combined with single-excitation configuration interaction (CIS) approach was employed to make a detailed investigation of the emitting energy for fifteen well-known coumarin derivatives. The results showed that the predicted emitting energies as well as the absorption ones were dominated mainly by the exchange-correlation (XC) functional to be used. So long as a functional is properly chosen, the experimental emitting energy of most derivatives can be accurately reproduced within 0.16 eV by a calculation at the TDDFT/6-31G(d)//CIS/3-21G(d) theoretical level. It was found that, nevertheless, the hybrid functional, B3LYP, well predicted the absorption energies for all the fifteen coumarin derivatives but none of the functionals could work equally well for the emitting energy calculations. Two pure functionals, OLYP and BLYP, yield good emitting energies for the 7-aminocoumarins or derivatives with a N atom connected to 7-position, which exhibit inconspicuous charge transfer (CT) in their excited states, whereas the B3LYP hybrid functional, with 20% Hartree-Fock (HF) exchange energy, performs significantly better than OLYP and BLYP for those 3-substituted coumarins with larger CT in excited states. Thus, in comparison with the absorption energies, the selection of proper functionals for the emitting energy calculations becomes more complex. In all probability, it is effective and doable to choose an XC-functional with alterable fraction of HF exchange energy according to the composition and structure characteristics of molecule.     

Electronic Structure Characteristics of ESIPT and TICT Fluorescence Emissions and Calculations of Emitting Energies

A scheme of time-dependent density functional theory (TDDFT) combined with the single-excitation configuration interaction (CIS) approach was employed to investigate the first excited singlet state (S₁) for eight salicylanilide derivatives and analogues, which have similar structural formulas. The results showed that fluorescence-emitting mechanisms of these molecules were in two distinct manners (excited-state intramolecular proton transfer (ESIPT) and twisted intramolecular charge transfer (TICT)), which agreed with the well-known experiments. For ESIPT compounds with inconspicuous charge transfer (CT) during electron transition, pure functionals without Hartree-Fock (HF) exchange energy, such as OLYP and BLYP, were suitable to calculate emitting energies. For TICT compounds with large CT during electron transition, hybrid functionals with about 37% HF exchange energy, such as mPW1B95 and MPW1K, performed well. On condition that the exchange-correlation (XC) functionals were chosen properly according to the rules above, reliable emitting energies for salicylanilide derivatives and analogues could be obtained at the TDDFT/6-31G(d)//CIS/3-21G(d) level. The average accuracy reached about 0.2 eV. For the salicylanilides with both proton transfer (PT) and CT reaction channels, only one channel occurred actually according to the principle of energy minimum. This actual reaction decided proper XC functionals, whereas the reaction that did not occur actually was trivial. Eight appendent compounds were calculated to prove that this successful scheme is expected to be suitable for other ESIPT and TICT compounds.     

ESIPT和TICT荧光发射的电子结构特征及发射能计算

采用含时密度泛函理论(TDDFT)与单激发组态相互作用(CIS)相结合的计算方案对八种结构相似的水杨酰苯胺衍生物及其类似物第一激发单重态(S1)进行考察, 证实它们的荧光发射分属分子内质子转移(ESIPT)和分子扭转-电荷转移(TICT)两种不同机制且结论与已知实验事实相符. ESIPT发光的化合物在电子跃迁前后无明显的电荷转移发生, 发射能计算的适用泛函是OLYP和BLYP等无Hartree-Fock(HF)交换成分的纯泛函; TICT发光的化合物在电子跃迁前后发生明显的电荷转移, 其适用泛函为含约37% HF交换成分的混合型泛函(例如mPW1B95和MPW1K). 按上述原则来选择适用泛函, 即可在TDDFT/6-31G(d)//CIS/3-21G(d)理论水平上正确预测水杨酰苯胺衍生物和类似物的发射能, 平均精度可达0.2 eV. 兼具质子转移与电荷转移双反应通道的化合物, 两者的竞争遵从能量最小原理, 结果使荧光发射仅选择其中一个通道进行. 泛函的选择只与实际发生的反应有关, 与并未实际发生的反应通道无关. 附加的八个算例进一步表明, 此成功的计算方案可望推广应用于其它类型的ESIPT和TICT荧光有机物.     

Excited-State Electronic Structure with Configuration Interaction Singles and Tamm-Dancoff Time-Dependent Density Functional Theory on Graphical Processing Units

Excited-state calculations are implemented in a development version of the GPU-based TeraChem software package using the configuration interaction singles (CIS) and adiabatic linear response Tamm-Dancoff time-dependent density functional theory (TDA-TDDFT) methods. The speedup of the CIS and TDDFT methods using GPU-based electron repulsion integrals and density functional quadrature integration allows full ab initio excited-state calculations on molecules of unprecedented size. CIS/6-31G and TD-BLYP/6-31G benchmark timings are presented for a range of systems, including four generations of oligothiophene dendrimers, photoactive yellow protein (PYP), and the PYP chromophore solvated with 900 quantum mechanical water molecules. The effects of double and single precision integration are discussed, and mixed precision GPU integration is shown to give extremely good numerical accuracy for both CIS and TDDFT excitation energies (excitation energies within 0.0005 eV of extended double precision CPU results).     

Efficient time-dependent density functional theory approximations for hybrid density functionals: analytical gradients and parallelization

In this paper, we present the implementation of efficient approximations to time-dependent density functional theory (TDDFT) within the Tamm-Dancoff approximation (TDA) for hybrid density functionals. For the calculation of the TDDFT/TDA excitation energies and analytical gradients, we combine the resolution of identity (RI-J) algorithm for the computation of the Coulomb terms and the recently introduced "chain of spheres exchange" (COSX) algorithm for the calculation of the exchange terms. It is shown that for extended basis sets, the RIJCOSX approximation leads to speedups of up to 2 orders of magnitude compared to traditional methods, as demonstrated for hydrocarbon chains. The accuracy of the adiabatic transition energies, excited state structures, and vibrational frequencies is assessed on a set of 27 excited states for 25 molecules with the configuration interaction singles and hybrid TDDFT/TDA methods using various basis sets. Compared to the canonical values, the typical error in transition energies is of the order of 0.01 eV. Similar to the ground-state results, excited state equilibrium geometries differ by less than 0.3 pm in the bond distances and 0.5° in the bond angles from the canonical values. The typical error in the calculated excited state normal coordinate displacements is of the order of 0.01, and relative error in the calculated excited state vibrational frequencies is less than 1%. The errors introduced by the RIJCOSX approximation are, thus, insignificant compared to the errors related to the approximate nature of the TDDFT methods and basis set truncation. For TDDFT/TDA energy and gradient calculations on Ag-TB2-helicate (156 atoms, 2732 basis functions), it is demonstrated that the COSX algorithm parallelizes almost perfectly (speedup ~26-29 for 30 processors). The exchange-correlation terms also parallelize well (speedup ~27-29 for 30 processors). The solution of the Z-vector equations shows a speedup of ~24 on 30 processors. The parallelization efficiency for the Coulomb terms can be somewhat smaller (speedup ~15-25 for 30 processors), but their contribution to the total calculation time is small. Thus, the parallel program completes a Becke3-Lee-Yang-Parr energy and gradient calculation on the Ag-TB2-helicate in less than 4 h on 30 processors. We also present the necessary extension of the Lagrangian formalism, which enables the calculation of the TDDFT excited state properties in the frozen-core approximation. The algorithms described in this work are implemented into the ORCA electronic structure system.     

Effect of hydration on the lowest singlet PiPi* excited-state geometry of guanine: a theoretical study

An ab-initio computational study was performed to investigate the effect of explicit hydration on the ground and lowest singlet PiPi* excited-state geometry and on the selected stretching vibrational frequencies corresponding to the different NH sites of the guanine acting as hydrogen-bond donors. The studied systems consisted of guanine interacting with one, three, five, six, and seven water molecules. Ground-state geometries were optimized at the HF level, while excited-state geometries were optimized at the CIS level. The 6-311G(d,p) basis set was used in all calculations. The nature of potential energy surfaces was ascertained via the harmonic vibrational frequency analysis; all structures were found minima at the respective potential energy surfaces. The changes in the geometry and the stretching vibrational frequencies of hydrogen-bond-donating sites of the guanine in the ground and excited state consequent to the hydration are discussed. It was found that the first solvation shell of the guanine can accommodate up to six water molecules. The addition of the another water molecule distorts the hydrogen-bonding network by displacing other neighboring water molecules away from the guanine plane.