Configuration interaction singles method with multilayer fragment molecular orbital scheme

Excitation configuration analysis for divide-and-conquer excited-state calculation method using dynamical polarizability

The authors previously developed a divide-and-conquer (DC)-based non-local excited-state calculation method for large systems using dynamical polarizability [Nakai and Yoshikawa, J. Chem. Phys. 146, 124123 (2017)]. This method evaluates the excitation energies and oscillator strengths using information on the dynamical polarizability poles. This article proposes a novel analysis of the previously developed method to obtain further configuration information on excited states, including excitation and de-excitation coefficients of each excitation configuration. Numerical applications to simple molecules, such as ethylene, hydrogen molecule, ammonia, and pyridazine, confirmed that the proposed analysis could accurately reproduce the excitation and de-excitation coefficients. The combination with the DC scheme enables both the local and non-local excited states of large systems with an excited nature to be treated.

Ab Initio Study of Charge Separation Dynamics and Pump-Probe Spectroscopy in the P3HT/PCBM Blend

We develop a bottom-up computational method for excited-state dynamics and time-resolved spectroscopy signals in molecular aggregates, on the basis of ab initio excited-state calculations. As an application, we consider the charge separation dynamics and pump-probe spectroscopy in the amorphous P3HT/PCBM blend. To simulate quantum dynamics and time-resolved spectroscopy, the model Hamiltonian for single-excitation and double-excitation manifolds was derived on the basis of fragment-based excited-state calculations within the GW approximation and the Bethe-Salpeter equation. After elucidating the energetics of the electron-hole separation and examining linear absorption spectrum, we investigated the quantum dynamics of exciton and charge carriers in comparison with the pump-probe transient absorption spectra. In particular, we introduced the pump-probe excited-state absorption (ESA) anisotropy as a spectroscopic signature of charge carrier dynamics after exciton dissociation. We found that the charge separation dynamics can be probed by the pump-probe ESA anisotropy dynamics after charge-transfer excitations. The present study provides the fundamental information for understanding the experimental spectroscopy signals, by elucidating the relationship between the excited states, the exciton and charge carrier dynamics, and time-resolved spectroscopy.

Analytic Gradient for Time-Dependent Density Functional Theory Combined with the Fragment Molecular Orbital Method

The analytic energy gradient of energy with respect to nuclear coordinates is derived for the fragment molecular orbital (FMO) method combined with time-dependent density functional theory (TDDFT). The response terms arising from the use of a polarizable embedding are derived. The obtained analytic FMO-TDDFT gradient is shown to be accurate in comparison to both numerical FMO-TDDFT and unfragmented TDDFT gradients, at the level of two- and three-body expansions. The gradients are used for geometry optimizations, molecular dynamics, vibrational calculations, and simulations of IR and Raman spectra of excited states. The developed method is used to optimize the geometry of the ground and excited electronic states of the photoactive yellow protein (PDB: 2PHY).

Long-range corrected fragment molecular orbital density functional tight-binding method for excited states in large molecular systems

Herein, we present a new method to efficiently calculate electronically excited states in large molecular assemblies, consisting of hundreds of molecules. For this purpose, we combine the long-range corrected tight-binding density functional fragment molecular orbital method (FMO-LC-DFTB) with an excitonic Hamiltonian, which is constructed in the basis of locally excited and charge-transfer configuration state functions calculated for embedded monomers and dimers and accounts explicitly for the electronic coupling between all types of excitons. We first evaluate both the accuracy and efficiency of our fragmentation approach for molecular dimers and aggregates by comparing it with the full LC-TD-DFTB method. The comparison of the calculated spectra of an anthracene cluster shows a very good agreement between our method and the LC-TD-DFTB reference. The effective computational scaling of our method has been explored for anthracene clusters and for perylene bisimide aggregates. We demonstrate the applicability of our method by the calculation of the excited state properties of pentacene crystal models consisting of up to 319 molecules. Furthermore, the participation ratio of the monomer fragments to the excited states is analyzed by the calculation of natural transition orbital participation numbers, which are verified by the hole and particle density for a chosen pentacene cluster. The use of our FMO-LC-TDDFTB method will allow for future studies of excitonic dynamics and charge transport to be performed on complex molecular systems consisting of thousands of atoms.

Divide-and-Conquer Linear-Scaling Quantum Chemical Computations

Fragmentation and embedding schemes are of great importance when applying quantum-chemical calculations to more complex and attractive targets. The divide-and-conquer (DC)-based quantum-chemical model is a fragmentation scheme that can be connected to embedding schemes. This feature article explains several DC-based schemes developed by the authors over the last two decades, which was inspired by the pioneering study of DC self-consistent field (SCF) method by Yang and Lee (J. Chem. Phys. 1995, 103, 5674-5678). First, the theoretical aspects of the DC-based SCF, electron correlation, excited-state, and nuclear orbital methods are described, followed by the two-component relativistic theory, quantum-mechanical molecular dynamics simulation, and the introduction of three programs, including DC-based schemes. Illustrative applications confirmed the accuracy and feasibility of the DC-based schemes.

A double exponential coupled cluster theory in the fragment molecular orbital framework

Fragmentation-based methods enable electronic structure calculations for large chemical systems through partitioning them into smaller fragments. Here, we have developed and benchmarked a dual exponential operator-based coupled cluster theory to account for high-rank electronic correlation of large chemical systems within the fragment molecular orbital (FMO) framework. Upon partitioning the molecular system into several fragments, the zeroth order reference determinants for each fragment and fragment pair are constructed in a self-consistent manner with two-body FMO expansion. The dynamical correlation is induced through a dual exponential ansatz with a set of fragment-specific rank-one and rank-two operators that act on the individual reference determinants. While the single and double excitations for each fragment are included through the conventional rank-one and rank-two cluster operators, the triple excitation space is spanned via the contraction between the cluster operators and a set of rank-two scattering operators over a few optimized fragment-specific occupied and virtual orbitals. Thus, the high-rank dynamical correlation effects within the FMO framework are computed with rank-one and rank-two parametrization of the wave operator, leading to significant reduction in the number of variables and associated computational scaling over the conventional methods. Through a series of pilot numerical applications on various covalent and non-covalently bonded systems, we have shown the quantitative accuracy of the proposed methodology compared to canonical, as well as FMO-based coupled-cluster single double triple. The accuracy of the proposed method is shown to be systematically improvable upon increasing the number of contractible occupied and virtual molecular orbitals employed to simulate triple excitations.

Local electronic structure analysis by ab initio elongation method: A benchmark using DNA block polymers

The ab initio elongation (ELG) method based on a polymerization concept is a feasible way to perform linear-scaling electronic structure calculations for huge aperiodic molecules while maintaining computational accuracy. In the method, the electronic structures are sequentially elongated by repeating (1) the conversion of canonical molecular orbitals (CMOs) to region-localized MOs (RLMOs), that is, active RLMOs localized onto a region close to an attacking monomer or frozen RLMOs localized onto the remaining region, and the subsequent (2) partial self-consistent-field calculations for an interaction space composed of the active RLMOs and the attacking monomer. For each ELG process, one can obtain local CMOs for the interaction space and the corresponding local orbital energies. Local site information, such as the local highest-occupied/lowest-unoccupied MOs, can be acquired with linear-scaling efficiency by correctly including electronic effects from the frozen region. In this study, we performed a local electronic structure analysis using the ELG method for various DNA block polymers with different sequential patterns. This benchmark aimed to confirm the effectiveness of the method toward the efficient detection of a singular local electronic structure in unknown systems as a future practical application. We discussed the high-throughput efficiency of our method and proposed a strategy to detect singular electronic structures by combining with a machine learning technique.

Towards a quantitative description of excitonic couplings in photosynthetic pigment-protein complexes: quantum chemistry driven multiscale approaches

A structure-based quantitative calculation of excitonic couplings between photosynthetic pigments has to describe the dynamical polarization of the protein/solvent environment of the pigments, giving rise to reaction field and screening effects. Here, this challenging problem is approached by combining the fragment molecular orbital (FMO) method with the polarizable continuum model (PCM). The method is applied to compute excitonic couplings between chlorophyll a (Chl a) pigments of the water-soluble chlorophyll-binding protein (WSCP). By calibrating the vacuum dipole strength of the 0–0 transition of the Chl a chromophores according to experimental data, an excellent agreement between calculated and experimental linear absorption and circular dichroism spectra of WSCP is obtained. The effect of the mutual polarization of the pigment ground states is calculated to be very small. The simple Poisson-Transition-charge-from-Electrostatic-potential (Poisson-TrEsp) method is found to accurately describe the screening part of the excitonic coupling, obtained with FMO/PCM. Taking into account that the reaction field effects of the latter method can be described by a scalar constant leads to an improvement of Poisson-TrEsp that is expected to provide the basis for simple and realistic calculations of optical spectra and energy transfer in photosynthetic light-harvesting complexes. In addition, we present an expression for the estimation of Huang–Rhys factors of high-frequency pigment vibrations from experimental fluorescence line-narrowing spectra that takes into account the redistribution of oscillator strength by the interpigment excitonic coupling. Application to WSCP results in corrected Huang–Rhys factors that are less than one third of the original values obtained by the standard electronic two-state analysis that neglects the above redistribution. These factors are important for the estimation of the dipole strength of the 0–0 transition of the chromophores and for the development of calculation schemes for the spectral density of the exciton-vibrational coupling.

The importance of reaction field and screening effects on the excitonic couplings is demonstrated, and from quantum-chemical calculations a single scaling factor is derived that can be used to improve simple models based on the Poisson equation.

Fragment-Based Excited-State Calculations Using the GW Approximation and the Bethe-Salpeter Equation

Herein, we present a fragment-based approach for calculating the charged and neutral excited states in molecular systems, based on the many-body Green's function method within the GW approximation and the Bethe-Salpeter equation (BSE). The implementation relies on the many-body expansion of the total irreducible polarizability on the basis of fragment molecular orbitals. The GW quasi-particle energies in complex molecular environments are obtained by the GW calculation for the target fragment plus induced polarization contributions of the surrounding fragments at the static Coulomb-hole plus screened exchange level. In addition, we develop a large-scale GW/BSE method for calculating the delocalized excited states of molecular aggregates, based on the fragment molecular orbital method and the exciton model. The accuracy of fragment-based GW and GW/BSE methods was evaluated on molecular clusters and molecular crystals. We found that the accuracy of the total irreducible polarizability can be improved systematically by including two-body correction terms, and the fragment-based calculations can reasonably reproduce the results of the corresponding unfragmented calculations with a relative error of less than 100 meV. The proposed approach enables efficient excited-state calculations for large molecular systems with reasonable accuracy.

Trajectory Surface Hopping Approach to Condensed-Phase Nonradiative Relaxation Dynamics Using Divide-and-Conquer Spin-Flip Time-Dependent Density-Functional Tight Binding

Nonradiative relaxation of excited molecules is central to many crucial issues in photochemistry. Condensed phases are typical contexts in which such problems are considered, and the nonradiative relaxation dynamics are expected to be significantly affected by interactions with the environment, for example, a solvent. We developed a nonadiabatic molecular dynamics simulation technique that can treat the nonradiative relaxation and explicitly include the environment in the calculations without a heavy computational burden. Specifically, we combined trajectory surface hopping with Tully's fewest-switches algorithm, a tight-binding approximated version of spin-flip time-dependent density-functional theory, and divide-and-conquer (DC) spatial fragmentation scheme. Numerical results showed that this method can treat systems with thousands of atoms within reasonable computational resources, and the error arising from DC fragmentation is negligibly small. Using this method, we obtained molecular insights into the solvent dependence of the photoexcited-state dynamics of trans-azobenzene, which demonstrate the importance of the environment for condensed-phase nonradiative relaxation.

Revisiting the Charge-Transfer States at Pentacene/C60 Interfaces with the GW/Bethe-Salpeter Equation Approach

Molecular orientations and interfacial morphologies have critical effects on the electronic states of donor/acceptor interfaces and thus on the performance of organic photovoltaic devices. In this study, we explore the energy levels and charge-transfer states at the organic donor/acceptor interfaces on the basis of the fragment-based GW and Bethe-Salpeter equation approach. The face-on and edge-on orientations of pentacene/C60 bilayer heterojunctions have employed as model systems. GW+Bethe-Salpeter equation calculations were performed for the local interface structures in the face-on and edge-on bilayer heterojunctions, which contain approximately 2000 atoms. Calculated energy levels and charge-transfer state absorption spectra are in reasonable agreements with those obtained from experimental measurements. We found that the dependence of the energy levels on interfacial morphology is predominantly determined by the electrostatic contribution of polarization energy, while the effects of induction contribution in the edge-on interface are similar to those in the face-on. Moreover, the delocalized charge-transfer states contribute to the main absorption peak in the edge-on interface, while the face-on interface features relatively localized charge-transfer states in the main absorption peak. The impact of the interfacial morphologies on the polarization and charge delocalization effects is analyzed in detail.

Large-scale excited-state calculation using dynamical polarizability evaluated by divide-and-conquer based coupled cluster linear response method

This study attempted to propose an efficient scheme at the coupled cluster linear response (CCLR) level to perform large-scale excited-state calculations of not only local excitations but also nonlocal ones such as charge transfers and transitions between delocalized orbitals. Although standard applications of fragmentation techniques to the excited-state calculations brought about the limitations that could only deal with local excitations, this study solved the problem by evaluating the excited states as the poles of dynamical polarizability. Because such an approach previously succeeded at the time-dependent density functional theory level [H. Nakai and T. Yoshikawa, J. Chem. Phys. 146, 124123 (2017)], this study was considered as an extension to the CCLR level. To evaluate the dynamical polarizability at the CCLR level, we revisited three equivalent formulas, namely, coupled-perturbed self-consistent field (CPSCF), random phase approximation (RPA), and Green's function (GF). We further extended these formulas to the linear-scaling methods based on the divide-and-conquer (DC) technique. We implemented the CCLR with singles and doubles (CCSDLR) program for the six schemes, i.e., the standard and DC-type CPSCF, RPA, and GF. Illustrative applications of the present methods demonstrated the accuracy and efficiency. Although the standard three treatments could exactly reproduced the conventional frequency-domain CCSDLR results, their computational costs were commonly higher than that of the conventional ones due to large amount of computations for individual frequencies of the external electric field. The DC-type treatments, which approximately reproduced the conventional results, could achieve quasilinear scaling computational costs. Among them, DC-GF was found to exhibit the best performance.

GPU-Accelerated Large-Scale Excited-State Simulation Based on Divide-and-Conquer Time-Dependent Density-Functional Tight-Binding

The present study implemented the divide-and-conquer time-dependent density-functional tight-binding (DC-TDDFTB) code on a graphical processing unit (GPU). The DC method, which is a linear-scaling scheme, divides a total system into several fragments. By separately solving local equations in individual fragments, the DC method could reduce slow central processing unit (CPU)-GPU memory access, as well as computational cost, and avoid shortfalls of GPU memory. Numerical applications confirmed that the present code on GPU significantly accelerated the TDDFTB calculations, while maintaining accuracy. Furthermore, the DC-TDDFTB simulation of 2-acetylindan-1,3-dione displays excited-state intramolecular proton transfer and provides reasonable absorption and fluorescence energies with the corresponding experimental values. © 2019 Wiley Periodicals, Inc.

Charge-transfer excited states in the donor/acceptor interface from large-scale GW calculations

Predicting the charge-transfer (CT) excited states across the donor/acceptor (D/A) interface is essential for understanding the charge photogeneration process in an organic solar cell. Here, we present a fragment-based GW implementation that can be applied to a D/A interface structure and thus enables accurate determination of the CT states. The implementation is based on the fragmentation approximation of the polarization function and the combined GW and Coulomb-hole plus screened exchange approximations for self-energies. The fragment-based GW is demonstrated by application to the pentacene/C₆₀ interface structure containing more than 2000 atoms. The CT excitation energies were estimated from the quasiparticle energies and electron-hole screened Coulomb interactions; the computed energies are in reasonable agreement with experimental estimates from the external quantum efficiency measurements. We highlight the impact of the induced polarization effects on the electron-hole energetics. The proposed fragment-based GW method offers a first-principles tool to compute the quasiparticle energies and electronic excitation energies of organic materials.

Thousand-atom ab initio calculations of excited states at organic/organic interfaces: toward first-principles investigations of charge photogeneration

Predicting electronically excited states across electron-donor/electron acceptor interfaces is essential for understanding the charge photogeneration process in organic solar cells. However, organic solar cells are large and disordered systems, and their excited states cannot be easily accessed by conventional quantum chemistry approaches. Moreover, a large number of excited states must be obtained to fully understand the charge separation mechanism. Recently, we have developed a novel fragment-based excited state method which can efficiently calculate a large number of states in molecular aggregates. In this article, we demonstrate the large-scale excited-state calculations by investigating interfacial charge transfer (ICT) states across the electron-donor/electron acceptor interfaces. As the model systems, we considered the face-on and edge-on configurations of pentacene/C₆₀ bilayer heterojunction structures. These model structures contain approximately 1.8 × 10⁵ atoms, and their local interface regions containing 2000 atoms were treated quantum mechanically, embedded in the electrostatic potentials from the remaining parts. Therefore, the charge delocalization effect, structural disorder, and the resulting heterogeneous electrostatic and polarizable environments were taken into account in the excited-state calculations. The computed energies of the low-lying ICT states are in reasonable agreement with experimental estimates. By comparing the edge-on and face-on configurations of the pentacene/C₆₀ interfaces, we discuss the influence of interfacial morphologies on the energetics and charge delocalization of ICT states. In addition, we present the detailed characterization of excited states and highlight the importance of hybridization effects between pentacene excited states and ICT states. The large-scale ab initio calculations for the interface systems enabled the exploration of the ICT states, leading to first-principles investigation of the charge separation mechanism in organic solar cells.

Coherent Dynamics of Mixed Frenkel and Charge-Transfer Excitons in Dinaphtho[2,3-b:2'3'-f]thieno[3,2-b]-thiophene Thin Films: The Importance of Hole Delocalization

Charge-transfer states in organic semiconductors play crucial roles in processes such as singlet fission and exciton dissociation at donor/acceptor interfaces. Recently, a time-resolved spectroscopy study of dinaphtho[2,3-b:2'3'-f]thieno[3,2-b]-thiophene (DNTT) thin films provided evidence for the formation of mixed Frenkel and charge-transfer excitons after the photoexcitation. Here, we investigate optical properties and excitation dynamics of the DNTT thin films by combining ab initio calculations and a stochastic Schrödinger equation. Our theory predicts that the low-energy Frenkel exciton band consists of 8-47% CT character. The quantum dynamics simulations show coherent dynamics of Frenkel and CT states in 50 fs after the optical excitation. We demonstrate the role of charge delocalization and localization in the mixing of CT states with Frenkel excitons as well as the role of their decoherence.

Electron-correlated fragment-molecular-orbital calculations for biomolecular and nano systems

Recent developments in the fragment molecular orbital (FMO) method for theoretical formulation, implementation, and application to nano and biomolecular systems are reviewed. The FMO method has enabled ab initio quantum-mechanical calculations for large molecular systems such as protein-ligand complexes at a reasonable computational cost in a parallelized way. There have been a wealth of application outcomes from the FMO method in the fields of biochemistry, medicinal chemistry and nanotechnology, in which the electron correlation effects play vital roles. With the aid of the advances in high-performance computing, the FMO method promises larger, faster, and more accurate simulations of biomolecular and related systems, including the descriptions of dynamical behaviors in solvent environments. The current status and future prospects of the FMO scheme are addressed in these contexts.

Efficient and accurate fragmentation methods

Conspectus Three novel fragmentation methods that are available in the electronic structure program GAMESS (general atomic and molecular electronic structure system) are discussed in this Account. The fragment molecular orbital (FMO) method can be combined with any electronic structure method to perform accurate calculations on large molecular species with no reliance on capping atoms or empirical parameters. The FMO method is highly scalable and can take advantage of massively parallel computer systems. For example, the method has been shown to scale nearly linearly on up to 131 000 processor cores for calculations on large water clusters. There have been many applications of the FMO method to large molecular clusters, to biomolecules (e.g., proteins), and to materials that are used as heterogeneous catalysts. The effective fragment potential (EFP) method is a model potential approach that is fully derived from first principles and has no empirically fitted parameters. Consequently, an EFP can be generated for any molecule by a simple preparatory GAMESS calculation. The EFP method provides accurate descriptions of all types of intermolecular interactions, including Coulombic interactions, polarization/induction, exchange repulsion, dispersion, and charge transfer. The EFP method has been applied successfully to the study of liquid water, π-stacking in substituted benzenes and in DNA base pairs, solvent effects on positive and negative ions, electronic spectra and dynamics, non-adiabatic phenomena in electronic excited states, and nonlinear excited state properties. The effective fragment molecular orbital (EFMO) method is a merger of the FMO and EFP methods, in which interfragment interactions are described by the EFP potential, rather than the less accurate electrostatic potential. The use of EFP in this manner facilitates the use of a smaller value for the distance cut-off (Rcut). Rcut determines the distance at which EFP interactions replace fully quantum mechanical calculations on fragment-fragment (dimer) interactions. The EFMO method is both more accurate and more computationally efficient than the most commonly used FMO implementation (FMO2), in which all dimers are explicitly included in the calculation. While the FMO2 method itself does not incorporate three-body interactions, such interactions are included in the EFMO method via the EFP self-consistent induction term. Several applications (ranging from clusters to proteins) of the three methods are discussed to demonstrate their efficacy. The EFMO method will be especially exciting once the analytic gradients have been completed, because this will allow geometry optimizations, the prediction of vibrational spectra, reaction path following, and molecular dynamics simulations using the method.

The use of many-body expansions and geometry optimizations in fragment-based methods

Conspectus Chemists routinely work with complex molecular systems: solutions, biochemical molecules, and amorphous and composite materials provide some typical examples. The questions one often asks are what are the driving forces for a chemical phenomenon? How reasonable are our views of chemical systems in terms of subunits, such as functional groups and individual molecules? How can one quantify the difference in physicochemical properties of functional units found in a different chemical environment? Are various effects on functional units in molecular systems additive? Can they be represented by pairwise potentials? Are there effects that cannot be represented in a simple picture of pairwise interactions? How can we obtain quantitative values for these effects? Many of these questions can be formulated in the language of many-body effects. They quantify the properties of subunits (fragments), referred to as one-body properties, pairwise interactions (two-body properties), couplings of two-body interactions described by three-body properties, and so on. By introducing the notion of fragments in the framework of quantum chemistry, one obtains two immense benefits: (a) chemists can finally relate to quantum chemistry, which now speaks their language, by discussing chemically interesting subunits and their interactions and (b) calculations become much faster due to a reduced computational scaling. For instance, the somewhat academic sounding question of the importance of three-body effects in water clusters is actually another way of asking how two hydrogen bonds affect each other, when they involve three water molecules. One aspect of this is the many-body charge transfer (CT), because the charge transfers in the two hydrogen bonds are coupled to each other (not independent). In this work, we provide a generalized view on the use of many-body expansions in fragment-based methods, focusing on the general aspects of the property expansion and a contraction of a many-body expansion in a formally two-body series, as exemplified in the development of the fragment molecular orbital (FMO) method. Fragment-based methods have been very successful in delivering the properties of fragments, as well as the fragment interactions, providing insights into complex chemical processes in large molecular systems. We briefly review geometry optimizations performed with fragment-based methods and present an efficient geometry optimization method based on the combination of FMO with molecular mechanics (MM), applied to the complex of a subunit of protein kinase 2 (CK2) with a ligand. FMO results are discussed in comparison with experimental and MM-optimized structures.

Theoretical characterization of excitation energy transfer in chlorosome light-harvesting antennae from green sulfur bacteria

We present a theoretical study of excitation dynamics in the chlorosome antenna complex of green photosynthetic bacteria based on a recently proposed model for the molecular assembly. Our model for the excitation energy transfer (EET) throughout the antenna combines a stochastic time propagation of the excitonic wave function with molecular dynamics simulations of the supramolecular structure and electronic structure calculations of the excited states. We characterized the optical properties of the chlorosome with absorption, circular dichroism and fluorescence polarization anisotropy decay spectra. The simulation results for the excitation dynamics reveal a detailed picture of the EET in the chlorosome. Coherent energy transfer is significant only for the first 50 fs after the initial excitation, and the wavelike motion of the exciton is completely damped at 100 fs. Characteristic time constants of incoherent energy transfer, subsequently, vary from 1 ps to several tens of ps. We assign the time scales of the EET to specific physical processes by comparing our results with the data obtained from time-resolved spectroscopy experiments.

The role of the exchange in the embedding electrostatic potential for the fragment molecular orbital method

We have examined the role of the exchange in describing the electrostatic potential in the fragment molecular orbital method and showed that it should be included in the total Fock matrix to obtain an accurate one-electron spectrum; however, adding it to the Fock matrices of individual fragments and pairs leads to very large errors. For the error analysis we have used the virial theorem; numerical tests have been performed for solvated phenol at the Hartree-Fock level with the 6-31G( *) and 6-311G( * *) basis sets.