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An adaptive potential energy surface generation method using curvilinear valence coordinates
Abstract
An automatic Born-Oppenheimer potential energy surface (PES) generation method AGAPES is presented designed for the calculation of vibrational spectra of large rigid and semi-rigid polyatomic molecules within the mid-infrared energy range. An adaptive approach guided by information from intermediate vibrational calculations in connection with a multi-mode expansion of the PES in internal valence coordinates is used and its versatility is tested for a selection of molecules: HNO, HClCO, and formaldoxime. Significant computational savings are reported. The possibility of linear scaling of the sampling grid size with the molecular size due to decrease of correlation of remote coordinates in large molecules is examined and finally, possible improvements are suggested.
... While the automated construction of PESs based on normal coordinates has reached some degree of sophistication, 24,87−90 the likewise construction based on internal coordinates as needed for molecules with large amplitude motions is technically less mature. 91 Progress in that direction accompanied by the automated identification of a meaningful or best coordinate system is highly desirable. ...
Advances in the development of quantum chemical methods and progress in multicore architectures in computer science made the simulation of infrared spectra of isolated molecules competitive with respect to established experimental methods. Although it is mainly the multidimensional potential energy surface that controls the accuracy of these calculations, the subsequent vibrational structure calculations need to be carefully converged in order to yield accurate results. As both aspects need to be considered in a balanced way, we focus on approaches for molecules of up to 12–15 atoms with respect to both parts, which have been automated to some extent so that they can be employed in routine applications. Alternatives to machine learning will be discussed, which appear to be attractive, as long as local regions of the potential energy surface are sufficient. The automatization of these methods is still in its infancy, and the generalization to molecules with large amplitude motions or molecular clusters is far from trivial, but many systems relevant for astrophysical studies are already in reach.
... Rectilinear coordinate systems are a convenient choice for general, large molecules and clusters, although generic curvilinear alternatives also exist. [108][109][110][111][112][113] The existence of negative potentials could perhaps be used as a means to assess the appropriateness of coordinate systems. This occurrence of negative potentials is not necessarily limited to rotors or other naturally curvilinear motions, however. ...
Simulations of anharmonic vibrational motion rely on computationally expedient representations of the governing potential energy surface. The n-mode representation (n-MR)—effectively a many-body expansion in the space of molecular vibrations—is a general and efficient approach that is often used for this purpose in vibrational self-consistent field (VSCF) calculations and correlated analogues thereof. In the present analysis, a lack of convergence in many VSCF calculations is shown to originate from negative and unbound potentials at truncated orders of the n-MR expansion. For cases of strong anharmonic coupling between modes, the n-MR can both dip below the true global minimum of the potential surface and lead to effective single-mode potentials in VSCF that do not correspond to bound vibrational problems, even for bound total potentials. The present analysis serves mainly as a pathology report of this issue. Furthermore, this insight into the origin of VSCF non-convergence provides a simple, albeit ad hoc, route to correct the problem by “painting in” the full representation of groups of modes that exhibit these negative potentials at little additional computational cost. Somewhat surprisingly, this approach also reasonably approximates the results of the next-higher n-MR order and identifies groups of modes with particularly strong coupling. The method is shown to identify and correct problematic triples of modes—and restore SCF convergence—in two-mode representations of challenging test systems, including the water dimer and trimer, as well as protonated tropine.
... For example, Aerts et al. 33 suggested positioning of grid points based on a local loss function combined with an energy filter to sample more points in the spectroscopic region of interest, namely, around the equilibrium geometry. The adaptive generation of adiabatic PES (AGAPES) approach of Richter et al. 34 uses a combination of an interpolation error and the average density of the vibrational wave function at certain points to decide their importance. Consequently, this approach uses a feedback of the quantity, which is supposed to be computed, for a proper and efficient construction of the PES. ...
The positions of grid points for representing a multidimensional potential energy surface (PES) have a non-negligible impact on its accuracy and the associated computational effort for its generation. Six different positioning schemes were studied for PESs represented by n-mode expansions as needed for the accurate calculation of anharmonic vibrational frequencies by means of vibrational configuration interaction theory. A static approach, which has successfully been used in many applications, and five adaptive schemes based on Gaussian process regression have been investigated with respect to the number of necessary grid points and the accuracy of the fundamental modes for a small set of test molecules. A comparison with a related, more sophisticated, and consistent approach by Christiansen et al. is provided. The impact of the positions of the ab initio grid points is discussed for multilevel PESs, for which the computational effort of the individual electronic structure calculations decreases for increasing orders of the n-mode expansion. As a result of that, the ultimate goal is not the maximal reduction of grid points but rather the computational cost, which is not directly related.
... The PES was fit in terms of internal valence coordinates 1 (i.e. bond stretches, angles and torsions) with the points generated by the Adaptive Generation of Adiabatic PES (AGAPES) procedure. 103 The coordinate definition used is depicted in Fig. 2. The barrier between the two minima along the torsional coordinate t 2 was found to be 4442 cm À1 , where the barrier height was found to be affected by the bond-length r 1 and the bending angle y 2 . The barrier of this PES is 20 cm À1 higher compared to the value reported by Tew and Mizukami. ...
In this article, we review recent first principles, anharmonic studies on the molecular vibrations of gaseous formic acid in its monomer form.
... The calculation of extended areas of potential energy surface (PES) for a polyatomic system by high level interaction configurations methods is a great challenge in quantum chemistry and needs very time-consuming resources. Different techniques and algorithms developed in our laboratory and reported in the literature [20,21] offer an accurate representation of PES. They remain, however, less accurate in the case of linear molecules. ...
The high-resolution infrared spectrum of IC3N has been the subject of numerous experimental studies. Relying on a hybrid anharmonic potential CCSD(T) -F12b/ MP2-F12 with cc-pVTZ-F12b basis sets and the application of a pure variational method (VCI), the IR spectrum of IC3N was calculated between 100 and 4600 cm⁻¹. These calculations allowed us to revisit the entire IR spectrum and assign a large part of its overtones, combinations bands with respect to experimental measurements. As it is shown in this work, the observed bands located at 1031 and 955 cm⁻¹ could be explained in terms of Fermi resonance.
... [1][2][3][4] The second issue is often alleviated with well-known direct dynamics approaches, which calculate SPs on the fly, or by using adaptive schemes for choosing grid points (for example, see Refs. [5][6][7]. However, these techniques alone are far from enough to achieve a linear-scaling behavior in the overall PES construction. ...
We present a combination of the recently developed double incremental expansion of potential energy surfaces with the well-established adaptive density-guided approach to grid construction. This unique methodology is based on the use of an incremental expansion for potential energy surfaces, known as n-mode expansion; an incremental many-body representation of the electronic energy; and an efficient vibrational density-guided approach to automated determination of grid dimensions and granularity. The reliability of the method is validated calculating potential energy surfaces and obtaining fundamental excitation energies for three moderate-size chain-like molecular systems. The use of our methodology leads to considerable computational savings for potential energy surface construction compared to standard approaches while maintaining a high level of accuracy in the resulting potential energy surfaces. Additional investigations indicate that our method can be applied to covalently bound and strongly interacting molecular systems, even though these cases are known to be very unfavorable for fragmentation schemes. We therefore conclude that the presented methodology is a robust and flexible approach to potential energy surface construction, which introduces considerable computational savings without compromising the accuracy of vibrational spectra calculations.
... The second issue is often alleviated with well-known direct dynamics approaches, which calculate SPs on the fly, or by using adaptive schemes for choosing grid points (for example, see Refs. [5][6][7]). However, these techniques alone are far from enough to achieve a linear-scaling behavior in the overall PES construction. ...
We present a combination of the recently developed double incremental expansion of potential energy surfaces with the well-established adaptive density-guided approach to grid construction. This unique methodology is based on the use of an incremental expansion for potential energy surfaces, known as n-mode expansion, an incremental many-body representation of the electronic energy, and an efficient vibrational density-guided approach to automated determination of grid dimensions and granularity. The reliability of the method is validated calculating potential energy surfaces and obtaining fundamental excitation energies for three moderate-size chain-like molecular systems. The results are compared to other approaches, which utilize static grid construction for supersystem and fragmentation calculation setups. The use of our methodology leads to considerable computational savings for potential energy surface construction and a major reduction in the number of required single point calculations can be achieved, while maintaining a high level of accuracy in the resulting potential energy surfaces. Additional investigations indicate that our method can be applied to covalently bound and strongly interacting molecular systems, even though these cases are known as being very unfavorable for fragmentation schemes. We therefore conclude that the presented methodology is a robust and flexible approach to potential energy surface construction, which introduces considerable computational savings without compromising the accuracy of vibrational spectra calculations.
... These points can be defined in terms of a fixed human-defined static grid. Iterative methods have also been successfully used to find efficient ways of sampling the PES, [20][21][22] including the adaptive density guided approach (ADGA), 20 which iteratively builds up a grid based on convergence of the vibrational density times ...
On the basis of a new extensive database constructed for the purpose, we assess various Machine Learning (ML) algorithms to predict energies in the framework of potential energy surface (PES) construction and discuss black box character, robustness, and efficiency. The database for training ML algorithms in energy predictions based on the molecular structure contains SCF, RI-MP2, RI-MP2-F12, and CCSD(F12*)(T) data for around 10.5 × 10⁶ configurations of 15 small molecules. The electronic energies as function of molecular structure are computed from both static and iteratively refined grids in the context of automized PES construction for anharmonic vibrational computations within the n-mode expansion. We explore the performance of a range of algorithms including Gaussian Process Regression (GPR), Kernel Ridge Regression, Support Vector Regression, and Neural Networks (NNs). We also explore methods related to GPR such as sparse Gaussian Process Regression, Gaussian process Markov Chains, and Sparse Gaussian Process Markov Chains. For NNs, we report some explorations of architecture, activation functions, and numerical settings. Different delta-learning strategies are considered, and the use of delta learning targeting CCSD(F12*)(T) predictions using, for example, RI-MP2 combined with machine learned CCSD(F12*)(T)-RI-MP2 differences is found to be an attractive option.
... The natural choice for describing such motions would be curvilinear coordinates but these come at the cost of rather complicated expressions for the kinetic energy operator, which require specialized methods in order to be treated correctly. [64][65][66][67][68][69][70] The treatment of the kinetic energy operator for curvilinear coordinates is in this sense a problem with a known solution but we chose to defer the implementation of this to future work. We note that local mode models have been successfully used for accurate representation of potentials, in particular, for heavy atom hydrogen stretches. ...
We present an approach to treat sets of general fit-basis functions in a single uniform framework, where the functional form is supplied on input, i.e., the use of different functions does not require new code to be written. The fit-basis functions can be used to carry out linear fits to the grid of single points, which are generated with an adaptive density-guided approach (ADGA). A non-linear conjugate gradient method is used to optimize non-linear parameters if such are present in the fit-basis functions. This means that a set of fit-basis functions with the same inherent shape as the potential cuts can be requested and no other choices with regards to the fit-basis functions need to be taken. The general fit-basis framework is explored in relation to anharmonic potentials for model systems, diatomic molecules, water, and imidazole. The behaviour and performance of Morse and double-well fit-basis functions are compared to that of polynomial fit-basis functions for unsymmetrical single-minimum and symmetrical double-well potentials. Furthermore, calculations for water and imidazole were carried out using both normal coordinates and hybrid optimized and localized coordinates (HOLCs). Our results suggest that choosing a suitable set of fit-basis functions can improve the stability of the fitting routine and the overall efficiency of potential construction by lowering the number of single point calculations required for the ADGA. It is possible to reduce the number of terms in the potential by choosing the Morse and double-well fit-basis functions. These effects are substantial for normal coordinates but become even more pronounced if HOLCs are used.
... Therefore, a PES generator, which uses a fixed coarse grained mesh of ab initio points with a subsequent interpolation to a fine grid, will inevitably lead to an inconsistent description of the PES unless the number of ab initio points is vast. Consequently, iterative procedures as the adaptive density-guided approach (ADGA) by Christiansen et al., 39 the adaptive generation of adiabatic PES (AGAPES) of Richter et al. 41 or our iterative interpolation scheme are necessary in order to obtain well-balanced PESs. 42 It is the interplay of the surface generator and the fast evaluation of preliminary vibrational wave functions within the iterations, which allow to control important parameters of the surface, e.g. the extension of the surface can be controlled by the density of the vibrational wave function. ...
This feature article discusses some selected aspects in the field of vibrational structure calculations based on vibrational self-consistent field, VSCF, and vibrational configuration interaction, VCI, theory. As the quality of such calculations depends strongly on the accuracy of the underlying multidimensional potential energy surface, PES, some techniques will be discussed to establish high-quality PESs in a fully automated manner. As an alternative to VCI theory multiconfiguration self-consistent field, VMCSCF, theory and in particular specific aspects concerning the integral evaluation relevant to both approaches will also be presented. Further aspects concern the efficient calculation of infrared intensities and Franck-Condon factors in vibronic transitions.(doi: 10.5562/cca2149)
... [92][93][94] When truncated at k < d, the representation is approximate, and it has been shown that vibrational levels computed with k 4 have errors of a few cm 21 . [95][96][97][98][99][100][101] The terms in each of the sums are called component functions. Instead of using an NN to directly fit a multidimensional PES, it is possible to use NNs to fit the component functions. ...
Development and applications of neural network (NN)-based approaches for representing potential energy surfaces (PES) of bound and reactive molecular systems are reviewed. Specifically, it is shown that when the density of ab initio points is low, NNs-based potentials with multibody or multimode structure are advantageous for representing high-dimensional PESs. Importantly, with an appropriate choice of the neuron activation function, PESs in the sum-of-products form are naturally obtained, thus addressing a bottleneck problem in quantum dynamics. The use of NN committees is also analyzed and it is shown that while they are able to reduce the fitting error, the reduction is limited by the nonrandom nature of the fitting error. The approaches described here are expected to be directly applicable in other areas of science and engineering where a functional form needs to be constructed in an unbiased way from sparse data. © 2014 Wiley Periodicals, Inc.
We find kinetic energy operator (KEO) models based on n -mode expansions to be flexible, systematically improvable and accurate KEO representations in vibrationally correlated calculations in curvilinear coordinate systems.
High-resolution vibrational spectra of C–H, O–H, and N–H stretches depend on both molecular conformation and environment as well as provide a window into the frequencies of many other vibrational degrees of freedom as a result of mode mixing. We review current theoretical strategies that are being deployed to both aid and guide the analysis of the data that are encoded in these spectra. The goal is to enhance the power of vibrational spectroscopy as a tool for probing conformational preferences, hydrogen bonding effects away from equilibrium, and energy flow pathways. Recent years have seen an explosion of new methods and strategies for solving the nuclear Schrödinger equation. Rather than attempt a comprehensive review, this work highlights specific molecular systems that we have chosen as representing bonding motifs that are important to chemistry and biology. We focus on the choices theoretical chemists make regarding the level of electronic structure theory, the representation of the potential energy surface, the selection of coordinates, preferences in basis sets, and methods of solution.
Expected final online publication date for the Annual Review of Physical Chemistry, Volume 74 is April 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
The intramolecular vibrational relaxation dynamics of formic acid and its deuterated isotopologues is simulated on the full-dimensional potential energy surface of Richter and Carbonnière [F. Richter and P. Carbonnière, J. Chem. Phys. 148, 064303 (2018)] using the Heidelberg MCTDH package. We focus on couplings with the torsion vibrational modes close to the trans- cis isomerisation coordinate from the dynamics of artificially excited vibrational mode overtones. The C-O stretch bright vibrational mode is coupled to the out-of-the plane torsion mode in HCOOH, where this coupling could be exploited for laser-induced trans-to- cis isomerisation. Strong isotopic effects are observed: deuteration of the hydroxyl group, i.e., in HCOOD and DCOOD, destroys the C-O stretch to torsion mode coupling whereas in DCOOH, little to no effect is observed.
The intramolecular vibrational relaxation dynamics of formic acid and its deuterated isotopologues is simulated on the full-dimensional potential energy surface of Richter and Carbonniere [F. Richter and P. Carbonniere, J. Chem. Phys., 148, 064303 (2018)] using the Heidelberg MCTDH package. Mode couplings with the torsion coordinate capturing most of the trans-cis isomerisation are identified in the dynamics of artificially excited vibrational mode overtones. The C-O stretch bright vibrational mode is coupled to the out-of-the plane torsion mode in HCOOH, where this coupling could be exploited for laser-induced trans-to-cis isomerisation. Strong isotopic effects are observed: deuteration of the hydroxyl group, i.e., in HCOOD and DCOOD, destroys the C-O stretch to torsion mode coupling whereas in DCOOH, little to no effect is observed.
A semi-automatic sampling and fitting procedure for generating sum-of-product (Born-Oppenheimer) potential energy surfaces based on a high-dimensional model representation is presented. The adaptive sampling procedure and subsequent fitting relies on energies only and can be used for re-fitting existing analytic potential energy surfaces in sum-of-product form or for direct fits from ab initio computa- tions. The method is tested by fitting ground electronic state potential energy surfaces for small to medium sized semi-rigid molecules, i.e., HFCO, HONO, and HCOOH, based upon ab initio computations at the CCSD(T)-F12/cc-pVTZ-F12 or MP2/aug-cc-pVTZ levels of theory. Vibrational eigenstates are computed using block improved relaxation in the Heidelberg MCTDH package and compared to available experimental and theoretical data. The new potential energy surfaces are compared to the best ones currently available for these molecules, in terms of accuracy, including of resulting vibrational states, required numbers of sampling points, and number of fitting parameters. The present procedure leads to compact expansions and scales well with the number of dimensions for simple potentials such as single or double wells.
A valence coordinate H 2 NOH ground state potential energy surface accurate for all levels up to 6000 cm ⁻¹ relative to trans zero point energy has been generated at the coupled-cluster single double triple-F12/aug-cc-pVTZ level encompassing the trans and cis as well as the N–H 2 permutational conformers. All cis and trans fundamentals and a complete set of eigenfunctions up to about 3100 cm ⁻¹ have been calculated and assigned using the improved relaxation method of the Heidelberg multi-configuration time-dependent Hartree package and an exact expression for the kinetic energy in valence coordinates generated by the TANA program. The average and maximal error to all observed transitions is about 6.3 and 14.6 cm ⁻¹ , respectively. Local cis eigenfunctions exist with up to two quanta in the isomerization mode ν 9 . Although no significant inversion splittings have been found up to the considered 3100 cm ⁻¹ , they are expected within the fundamental energy range in view of the calculated 4261 cm ⁻¹ H 2 permutation/inversion barrier height. The cis-NH 2 symmetric stretch fundamental shows a Fermi resonance with a splitting of about 10 cm ⁻¹ .
The sum-of-products finite-basis-representation (SOP-FBR) approach for the automated multidimensional fit of potential energy surfaces (PESs) is presented. In its current implementation, the method yields a PES in the so-called Tucker sum-of-products form, but it is not restricted to this specific ansatz. The novelty of our algorithm lies in the fact that the fit is performed in terms of a direct product of a Schmidt basis, also known as natural potentials. These encode in a non-trivial way all the physics of the problem and, hence, circumvent the usual extra ad hoc and a posteriori adjustments (e.g., damping functions) of the fitted PES. Moreover, we avoid the intermediate refitting stage common to other tensor-decomposition methods, typically used in the context of nuclear quantum dynamics. The resulting SOP-FBR PES is analytical and differentiable ad infinitum. Our ansatz is fully general and can be used in combination with most (molecular) dynamics codes. In particular, it has been interfaced and extensively tested with the Heidelberg implementation of the multiconfiguration time-dependent Hartree quantum dynamical software package.
The high-resolution infrared spectrum of HC3N has been the subject of numerous experimental studies. However, none of these studies has been based on high-quality theoretical work. On the basis of an anharmonic potential CCSD(T)-F 12 /aug-cc-pVTZ and the use of a pure variational method (VCI) the IR spectrum of HC3N was calculated between 200 and 4800 cm⁻¹. These calculations make it possible, on the basis of the positioning of the bands and the calculation of their intensity, to revisit the entire IR spectrum and to assign a large part of it, overtones, combinations bands and hot bands to the experimental observations.
The vibrational eigenenergies of the deuterated forms of formic acid (DCOOD, HCOOD, and DCOOH) have been computed using the block-improved relaxation method, as implemented in the Heidelberg multiconfiguration time-dependent Hartree package on a previously published potential energy surface [F. Richter and P. Carbonnière, J. Chem. Phys. 148, 064303 (2018)] generated at the CCSD(T)-F12a/aug-cc-pVTZ-F12 level of theory. Fundamental, combination band, and overtone transition frequencies of the trans isomer were computed up to ∼3000 cm⁻¹ with respect to the zero point energy, and assignments were determined by visualization of the reduced densities. Root mean square deviations of computed fundamental transition frequencies with experimentally available gas-phase measurements are 8, 7, and 3 cm⁻¹ for trans-DCOOD, trans-HCOOD, and trans-DCOOH, respectively. The fundamental transition frequencies are provided for the cis isomer of all deuterated forms; experimental measurements of these frequencies for the deuterated cis isotopologues are not yet available, and the present work may guide their identification.
On the grounds of a hybrid CCSD(T)/B3LYP/aug-cc-pVTZ anharmonic potential and the use of a variational and variational-perturbational methods, the IR spectra of dicyanodiacetylene C 6 N 2 is revisited in the mid-infrared region up to 4500 cm ⁻¹ . A position and intensity analysis of our theoretical results allow us to assign the fundamental bands together with their combinations and overtones, in the aforementioned range of frequencies. The pure theoretical results are discussed in relation with the pure experimental data. The main objective of this paper is to give an ‘a priori’ complete IR spectrum of C 6 N 2 which can be used as a guide of the low-intensity bands in areas not completely assigned so far.
We present significant algorithmic improvements to a recently proposed direct quantum dynamics method, based upon combining well established grid-based quantum dynamics approaches and expansions of the potential energy operator in terms of a weighted sum of Gaussian functions. Specifically, using a sum of low-dimensional Gaussian functions to represent the potential energy surface (PES), combined with a secondary fitting of the PES using singular value decomposition, we show how standard grid-based quantum dynamics methods can be dramatically accelerated without loss of accuracy. This is demonstrated by on-the-fly simulations (using both standard grid-based methods and multi-configuration time-dependent Hartree) of both proton transfer on the electronic ground state of salicylaldimine and the non-adiabatic dynamics of pyrazine.
One single full dimensional valence coordinate HCOOH ground state potential energy surface accurate for both cis and trans conformers for all levels up to 6000 cm⁻¹ relative to trans zero point energy has been generated at CCSD(T)-F12a/aug-cc-pVTZ level. The fundamentals and a set of eigenfunctions complete up to about 3120 and 2660 cm⁻¹ for trans- and cis-HCOOH, respectively, have been calculated and assigned using the improved relaxation method of the Heidelberg multi-configuration time-dependent Hartree package and an exact expression for the kinetic energy in valence coordinates generated by the TANA program. The calculated trans fundamental transition frequencies agree with experiment to within 5 cm⁻¹. A few reassignments are suggested. Our results discard any cis trans delocalization effects for vibrational eigenfunctions up to 3640 cm⁻¹ relative to trans zero point energy.
On the grounds of a hybrid CCSD(T)/B3LYP/aug-cc-pVTZ anharmonic potential and the use of a variational and variational-perturbational methods, the IR spectra of 5-bromo-2,4-pentadiynenitrile is revisited in the mid-infrared region up to 4500 cm-1. A position and intensity analysis of our theoretical results allow us to assign the fundamental bands together with their combinations and overtones, in the aforementioned range of frequencies. The main objective of this paper is to give an 'a priori' complete IR spectrum of BrC5N which can be used as a guide of the low-intensity bands in areas not completely studied so far.
We extend the fragmentation-based double incremental expansion in FALCON coordinates (DIF) and its linear-scaling analogue [König and Christiansen, J. Chem. Phys., 2016, 145, 064105] to dipole surfaces. Thereby, we enable the calculation of intensities in vibrational absorption spectra from these cost-efficient property surfaces. We validate the obtained potential energy and dipole surfaces by vibrational spectra calculations employing damped response theory for correlated vibrational coupled cluster wave functions. Our largest calculation on a hexa-phenyl includes all 180 vibrational degrees of freedom of the system, which illustrates the potential of both the DIF schemes for property surface generation and the use of damped response theory from high-dimensional correlated vibrational wave functions. Generally, we obtain good agreement between the spectra calculated from the DIF property surfaces and the non-fragmented analogues. Moreover, when adopting suitable electronic structure methods, good agreement with respect to the experiment can be obtained, as shown for the example of 5-methylfurfural and RI-MP2. In conclusion, our results illustrate that the presented scheme with linearly scaling surfaces enables high quality spectra, as long as reasonably sized fragments can be defined. With this work, we push the realistic limits of vibrational spectra calculations from vibrational wave function methods and accurate electronic structure calculations to significantly larger systems than currently accessible.
We present a combination of the incremental expansion of potential energy surfaces (PESs), known as n-mode expansion, with the incremental evaluation of the electronic energy in a many-body approach. The application of semi-local coordinates in this context allows the generation of PESs in a very cost-efficient way. For this, we employ the recently introduced FALCON (Flexible Adaptation of Local COordinates of Nuclei) coordinates. By introducing an additional transformation step, concerning only a fraction of the vibrational degrees of freedom, we can achieve linear scaling of the accumulated cost of the single point calculations required in the PES generation. Numerical examples of these double incremental approaches for oligo-phenyl examples show fast convergence with respect to the maximum number of simultaneously treated fragments and only a modest error introduced by the additional transformation step. The approach, presented here, represents a major step towards the applicability of vibrational wave function methods to sizable, covalently bound systems.
Highly accurate multi-dimensional potential energy surfaces have been computed in a fully automated fashion using newly implemented grid computing capabilities, which allow for the use of an unlimited number of cores. This new feature, which has been interfaced to our potential energy surface generator, allows for the accurate investigation of molecular systems, which are significantly larger than reported in the recent literature. Multi-dimensional potential energy surfaces at the coupled-cluster level were generated for systems of up to 16 atoms, which were used to compute accurate anharmonic vibrational spectra, which can directly be compared with experimental data.
A highly correlated approach using curvilinear valence coordinates is applied to calculate the vibrational fundamentals and some combination modes of the formamide molecule with high accuracy. A series of potential energy surfaces (PESs) has been generated by AGAPES - a program for adaptive generation of adiabatic PESs - at various electronic structure qualities until excellent non-accidental agreement with the experimentally assigned fundamental transitions was reached at the CCSDT(T)-F12a/aug-cc-pVTZ level of theory using the improved relaxation method of the Heidelberg multi-configuration time-dependent Hartree (MCTDH) package in connection with an exact expression for the kinetic energy in valence coordinates generated by the TANA program. By comparison of the overtone series ν1-3ν1 to experiment we demonstrate that the known problems concerning the floppy ν1 wagging motion are solved within this approach. The potential energy coupling as well as the vibrational coupling in curvilinear coordinates is discussed together with the efficiency of this approach.
Die Berechnung von Tunnelaufspaltungen in molekularen Schwingungsspektren erfordert die genaue Berechnung von Potentialenergiehyperflächen. Dies ist für größere Moleküle ein sehr rechenzeitintensiver Schritt. Es existiert eine Vielzahl an Möglichkeiten zur Reduktion der benötigten Rechenzeit, wie zum Beispiel das Ausnutzen der Molekülsymmetrie, Multi-Level-Ansätze sowie iterative 'Fitting'-Verfahren. Außerdem stehen massiv parallele Implementierungen, darunter auch ein 'Gridcomputing Interface' zur Verfügung. Die maximalen Auslenkungen für die Potentialenergiehyperfläche können durch ein automatisiertes, auf der eindimensionalen Schwingungsdichte basierendes Verfahren, bestimmt werden.
Mit der so voll automatisch erzeugten Potentialenergiehyperfläche kann die Energie von Schwingungszuständen und damit auch Tunnelaufspaltungen bestimmt werden. Dazu existiert eine Vielzahl an Schwingungsstrukturverfahren, wie zum Beispiel der VSCF-Theorie mit anschließendem Schwingungskonfigurationswechselwirkungsverfahren. Allerdings erfordern einige Moleküle, wie zum Beipiel Ammoniak, auch eine akurate Beschreibung der Schwingungsdrehimpulsterme. Eine Reihe von Benchmarkrechnungen wird vorgestellt.
The vibrational spectra of a series of small lithium fluoride clusters, i.e. (LiF)n, n = 2-10, were studied by vibrational configuration interaction (VCI) calculations relying on potential energy surfaces including three-mode coupling terms and being obtained from explicitly correlated local coupled cluster calculations. Due to the account for anharmonicity effects, the simulated spectra allow for a direct comparison with experimental data and may thus help to identify clusters in the experiments. Even structurally closely related clusters can clearly be distinguished by infrared spectroscopy. The largest system in this study required more than 1000 basis functions in the electronic structure calculations and more than 10(7) configurations in the vibrational structure calculations and became computationally feasible only due to a combination of different approximations and highly parallelized algorithms.
We investigate the possible locality of potential energy surface (PES) coupling in curvilinear internal valence coordinates using pure electronic energy as well as vibrational energy guided definitions of PES coupling range on the example of the floppy but-enal and the semirigid but-dienol molecule C4OH6. We propose ways to exploit found coupling range limits for efficient PES generation leading to significant computational savings. The generation of the 27 dimensional PESs using vibrationally guided convergence criteria within an adaptive PES generation method (AGAPES) at B3LYP quality is reported as well with a detailed error-gain analysis. © 2014 Wiley Periodicals, Inc.
The vibrational Hamiltonian involves two high dimensional operators, the kinetic energy operator (KEO), and the potential energy surface (PES). Both must be approximated for systems involving more than a few atoms. Adaptive approximation schemes are not only superior to truncated Taylor or many-body expansions (MBE), they also allow for error estimates, and thus operators of predefined precision. To this end, modified sparse grids (SG) are developed that can be combined with adaptive MBEs. This MBE/SG hybrid approach yields a unified, fully adaptive representation of the KEO and the PES. Refinement criteria, based on the vibrational self-consistent field (VSCF) and vibrational configuration interaction (VCI) methods, are presented. The combination of the adaptive MBE/SG approach and the VSCF plus VCI methods yields a black box like procedure to compute accurate vibrational spectra. This is demonstrated on a test set of molecules, comprising water, formaldehyde, methanimine, and ethylene. The test set is first employed to prove convergence for semi-empirical PM3-PESs and subsequently to compute accurate vibrational spectra from CCSD(T)-PESs that agree well with experimental values.
The recently proposed scheme for representing multidimensional potential energy surfaces as a linear combination of products of one‐dimensional functions is extended. The extensions prove to be important if one proceeds to higher dimensions. An iteration procedure is introduced which can further improve the representation. The product representation of potential energy surfaces is especially well suited to be employed within the framework of the multiconfiguration time‐dependent Hartree (MCTDH) approximation. The potential representation scheme cannot only be used to represent given analytical potential energy surfaces, but also to interpolate multidimensional surfaces on given, e.g. ab initio, product grid points. The product representation method is applied to the three‐dimensional S1 electronic surface of NOCl and to a six‐dimensional model Coulomb potential. To check the quality of the NOCl surface representation, the photoabsorption spectrum for an excitation from the S0 to the S1 surface is computed. Weight functions are shown to be easily implemented and, in the case of the NOCl surface, allow a substantial reduction of the number of required expansion coefficients. Exploiting the underlying symmetries of the potential under consideration can further reduce the computational effort, as is shown in the example of the Coulomb potential. Finally, the NOCl S1 potential surface defined on 616 ab initio points is interpolated, as an example for the product interpolation scheme. © 1996 American Institute of Physics.
An algorithm for first-principles calculation of vibrational spectroscopy of polyatomic molecules is proposed, which combines electronic ab initio codes with the vibrational self-consistent field (VSCF) method, and with a perturbation-theoretic extension of VSCF. The integrated method directly uses points on the potential energy surface, computed from the electronic ab initio code, in the VSCF part. No fitting of an analytic potential function is involved. A key element in the approach is the approximation that only interactions between pairs of normal modes are important, while interactions of triples or more can be neglected. This assumption was found to hold well in applications. The new algorithm was applied to the fundamental vibrational excitations of H2O, Cl−(H2O), and (H2O)2, using the Möller–Plesset method for the electronic structure. The vibrational frequencies found are in very good accord with experiments. Estimates suggest that this electronic ab initio/VSCF approach should be feasible, with reasonable computational resources, for all-mode calculations of vibrational energies and wave functions for systems of up to 10–15 atoms. The new method can be also very useful for testing the accuracy of electronic structure codes by comparing with experimental vibrational spectroscopy. © 1999 American Institute of Physics.
In this article, we review state-of-the-art methods for computing vibrational energies of polyatomic molecules using quantum mechanical, variationally-based approaches. We illustrate the power of those methods by presenting applications to molecules with more than four atoms. This demonstrates the great progress that has been made in this field in the last decade in dealing with the exponential scaling with the number of vibrational degrees of freedom. In this review we present three methods that effectively obviate this bottleneck. The first important idea is the n-mode representation of the Hamiltonian and notably the potential. The potential (and other functions) is represented as a sum of terms that depend on a subset of the coordinates. This makes it possible to compute matrix elements, form a Hamiltonian matrix, and compute its eigenvalues and eigenfunctions. Another approach takes advantage of this multimode representation and represents the terms as a sum of products. It then exploits the powerful multiconfiguration Hartree time-dependent method to solve the time-dependent Schroinger equation and extract the eigenvalue spectrum. The third approach we present uses contracted basis functions in conjunction with a Lanczos eigensolver. Matrix vector products are done without transforming to a direct-product grid. The usefulness of these methods is demonstrated for several example molecules, e.g. methane, methanol and the Zundel cation.
We consider exact and approximate multivariate interpolation of a function f(x
1 , . . . , x
d
) by a rational function p
n,m
/q
n,m
(x
1 , . . . , x
d
) and develop an error formula for the difference f − p
n,m
/q
n,m
. The similarity with a well-known univariate formula for the error in rational interpolation is striking. Exact interpolation
is through point values for f and approximate interpolation is through intervals bounding f. The latter allows for some measurement error on the function values, which is controlled and limited by the nature of the
interval data. To achieve this result we make use of an error formula obtained for multivariate polynomial interpolation,
which we first present in a more general form. The practical usefulness of the error formula in multivariate rational interpolation
is illustrated by means of a 4-dimensional example, which is only one of the several problems we tested it on.
KeywordsMultivariate interpolation-Rational interpolation-Interpolation error
We report full-dimensional, ab initio potential energy and dipole moment surfaces, denoted PES and DMS, respectively, for arbitrary numbers of water monomers. The PES is a sum of 1-, 2-, and 3-body potentials which can also be augmented by semiempirical long-range higher-body interactions. The 1-body potential is a spectroscopically accurate monomer potential, and the 2- and 3-body potentials are permutationally invariant fits to tens of thousands of CCSD(T)/aug-cc-pVTZ and MP2/aug-cc-pVTZ electronic energies, respectively. The DMS is a sum of 1- and 2-body DMS, which are covariant fits to tens of thousands MP2/aug-cc-pVTZ dipole moment data. We present the details of these new 2- and 3-body potentials and then extensive applications and tests of this PES are made to the structures, classical binding energies, and harmonic frequencies of water clusters up to the 22-mer. In addition, we report the dipole moment for these clusters at various minima and compare the results against available and new ab initio calculations.
Global analytic potential energy surfaces for O((3)P) + H(2)O((1)A(1)) collisions, including the OH + OH hydrogen abstraction and H + OOH hydrogen elimination channels, are presented. Ab initio electronic structure calculations were performed at the CASSCF + MP2 level with an O(4s3p2d1f)/H(3s2p) one electron basis set. Approximately 10(5) geometries were used to fit the three lowest triplet adiabatic states corresponding to the triply degenerate O((3)P) + H(2)O((1)A(1)) reactants. Transition state theory rate constant and total cross section calculations using classical trajectories to collision energies up to 120 kcal mol(-1) (∼11 km s(-1) collision velocity) were performed and show good agreement with experimental data. Flux-velocity contour maps are presented at selected energies for H(2)O collisional excitation, OH + OH, and H + OOH channels to further investigate the dynamics, especially the competition and distinct dynamics of the two reactive channels. There are large differences in the contributions of each of the triplet surfaces to the reactive channels, especially at higher energies. The present surfaces should support quantitative modeling of O((3)P) + H(2)O((1)A(1)) collision processes up to ∼150 kcal mol(-1).
Full-dimensional ab initio potential energy surface (PES) and dipole moment surface (DMS) are reported for H(5)O(2) (+). Tens of thousands of coupled-cluster [CCSD(T)] and second-order Moller-Plesset (MP2) calculations of electronic energies, using aug-cc-pVTZ basis, were done. The energies were fit very precisely in terms of all the internuclear distances, using standard least-square procedures, however, with a fitting basis that satisfies permutational symmetry with respect to like atoms. The H(5)O(2) (+) PES is a fit to 48 189 CCSD(T) energies, containing 7962 polynomial coefficients. The PES has a rms fitting error of 34.9 cm(-1) for the entire data set up to 110 000 cm(-1). This surface can describe various internal floppy motions, including the H atom exchanges, monomer inversions, and monomer torsions. First- and higher-order saddle points have been located on the surface and compared with available previous theoretical work. In addition, the PES dissociates correctly (and symmetrically) to H(2)O+H(3)O(+), with D(e)=11 923.8 cm(-1). Geometrical and vibrational properties of the monomer fragments are presented. The corresponding global DMS fit (MP2 based) involves 3844 polynomial coefficients and also dissociates correctly.
We combine the high dimensional model representation (HDMR) idea of Rabitz and co-workers [J. Phys. Chem. 110, 2474 (2006)] with neural network (NN) fits to obtain an effective means of building multidimensional potentials. We verify that it is possible to determine an accurate many-dimensional potential by doing low dimensional fits. The final potential is a sum of terms each of which depends on a subset of the coordinates. This form facilitates quantum dynamics calculations. We use NNs to represent HDMR component functions that minimize error mode term by mode term. This NN procedure makes it possible to construct high-order component functions which in turn enable us to determine a good potential. It is shown that the number of available potential points determines the order of the HDMR which should be used.
By using exponential activation functions with a neural network (NN) method we show that it is possible to fit potentials to a sum-of-products form. The sum-of-products form is desirable because it reduces the cost of doing the quadratures required for quantum dynamics calculations. It also greatly facilitates the use of the multiconfiguration time dependent Hartree method. Unlike potfit product representation algorithm, the new NN approach does not require using a grid of points. It also produces sum-of-products potentials with fewer terms. As the number of dimensions is increased, we expect the advantages of the exponential NN idea to become more significant.
We propose a method for fitting potential energy surfaces with a sum of component functions of lower dimensionality. This form facilitates quantum dynamics calculations. We show that it is possible to reduce the dimensionality of the component functions by introducing new and redundant coordinates obtained with linear transformations. The transformations are obtained from a neural network. Different coordinates are used for different component functions and the new coordinates are determined as the potential is fitted. The quality of the fits and the generality of the method are illustrated by fitting reference potential surfaces of hydrogen peroxide and of the reaction OH+H(2)-->H(2)O+H.
We report a new full-dimensional potential energy surface (PES) for the water dimer, based on fitting energies at roughly 30,000 configurations obtained with the coupled-cluster single and double, and perturbative treatment of triple excitations method using an augmented, correlation consistent, polarized triple zeta basis set. A global dipole moment surface based on Moller-Plesset perturbation theory results at these configurations is also reported. The PES is used in rigorous quantum calculations of intermolecular vibrational frequencies, tunneling splittings, and rotational constants for (H2O)2 and (D2O)2, using the rigid monomer approximation. Agreement with experiment is excellent and is at the highest level reported to date. The validity of this approximation is examined by comparing tunneling barriers within that model with those from fully relaxed calculations.
We report vibrational configuration interaction calculations of the monomer fundamentals of (H(2)O)(2), (D(2)O)(2), (H(2)O)(3), and (D(2)O)(3) using the code MULTIMODE and full dimensional ab initio-based global potential energies surfaces (PESs). For the dimer the HBB PES [Huang et al., J. Chem. Phys 128, 034312 (2008)] is used and for the trimer a new PES, reported here, is used. The salient properties of the new trimer PES are presented and compared to previous single-point calculations and the vibrational energies are compared with experiments.
This review focuses on the calculation of rovibrational energies of polyatomic molecules using the code MULTIMODE. This code, which uses normal coordinates and a hierarchical n-mode representation of the potential, aims to be applicable to a wide class of molecules and molecular complexes. The theoretical and computational methods used in this code are described, followed by a review of selected applications. These applications illustrate various features of the code and also point out some limitations of the current version of the code. The review concludes with some ideas about possible future directions in this area of research.
It is shown that the description of a three-atom molecular system by two valence relative position vectors is, within the framework of an adequate representation previously introduced for Jacobi vectors, also advantageous with regard to the criterion of maximal prediagonalization of the matrix representing the kinetic energy operator.
A molecular potential energy surface has the symmetry properties of invariance to rotation of the whole molecule, inversion of all atomic coordinates, and permutation of indistinguishable nuclei. While some of this invariance character can be easily incorporated in a local description of the surface, a formal application of these symmetry restrictions is useful in considering the form of the global surface which must account for large amplitude changes of the atomic coordinates. The form of a global molecular potential energy surface as a properly symmetrized analytic function of Cartesian coordinates is derived by extending Molien’s theorem of invariants for finite groups to cover the continuous rotation–inversion group. O(3), and the product of O(3) with the complete nuclear permutation group. The role of so‐called redundant internal coordinates in molecular potential energy surfaces is clarified.
The efficiency of the theoretical methods to obtain exact expressions of classical and quantum-mechanical Hamiltonians for N-particle systems described in terms of curvilinear coordinates (q), is shown to depend strongly on whether the system is subjected to constraints or not. If it is free, the method based upon the use of the relations q = q(x), where x denotes moving-frame Cartesian coordinates of the particles, is preferable. If it is constrained, the method making use of x = x(q) can become more efficient.
Whereas model constraints (namely, internal degrees of freedom either frozen or stepwise adjusted by gradient methods) are often imposed for calculating the potential energies of polyatomic molecules by quantum-chemical methods, the derivation of exact expressions for the corresponding kinetic energy operators is difficult because of the changes of metrics of the configuration spaces, which modify the differential operators but not the multiplicative operators. An appropriate method for overcoming this difficulty has been designed in the case of rigid constraints (e.g., frozen groups) (M. Menou and X. Chapuisat,J. Mol. Spectrosc.159,300–328, 1993). In this article, it is generalized to the case of adiabatic constraints; i.e., the variations of certain internal degrees of freedom are adjusted to those of other degrees of freedom. Exact kinetic energy operators are derived. An example is analyzed.
A generic high dimensional model representation (HDMR) method is presented for approximating multivariate functions in terms of functions of fewer variables and for going beyond the tensor-product formulation. Within the framework of reproducing kernel Hilbert space (RKHS) interpolation techniques, an HDMR is formulated for constructing global potential energy surfaces. The HDMR tools in conjunction with a successive multilevel decomposition technique provide efficient and accurate procedures for reducing a multidimensional interpolation problem to smaller, independent subproblems. It is shown that, when compared to the conventional tensor-product approach, the RKHS–HDMR methods can accurately produce smooth potential energy surfaces over dynamically relevant, nonrectangular regions using far fewer ab initio data points. Numerical results are given for a reduced two-level RKHS–HDMR of the C(1D)+H2 reactive system. The proposed RKHS–HDMR is intimately related to Gordon’s blending-function methods for multivariate interpolation and approximation. The general findings in the paper and the successful illustration provide a foundation for further applications of the techniques. © 2003 American Institute of Physics.
This paper aims at presenting a general and compact matrix expression of the exact kinetic energy operator in polyspherical coordinates adapted to the study of semirigid molecules. The internal coordinates of an N atom system are expressed by a set of N−1 relative position vectors. The operator can be applied to whatever the set of vectors (Jacobi, Radau, valence, satellite, etc., or a combination of these vectors), and whatever the number of atoms. It includes the rotation and the Coriolis coupling. Such a formulation gives the opportunity to develop a general code for calculating the ro-vibrational spectra in a curvilinear description including all the vibrational, rotational, and Coriolis couplings. © 2001 American Institute of Physics.
We report calculations of the vibrational energies of CO–Cu(100) using a new code to perform vibrational self-consistent field (VSCF) and state-mixing calculations for many-mode systems. The major new feature of the code is the representation of the potential. Unlike recent implementations of the VSCF method, the potential is not expanded in terms of normal coordinates as a multinomial series about a minimum. The full potential, in normal coordinates, is used in the Watson Hamiltonian. This approach, while rigorous, can lead to prohibitively large numerical quadratures, and so we suggest a novel representation of the potential as an expansion in all two-mode, or all three-mode, or all four-mode coupling terms. The new code is tested against previous exact calculations of vibrational states of HCO, and also against previous VSCF calculations that used a fourth-order, normal coordinate force field representation of the global HCO potential. The new code is applied to calculations of the vibrations of CO adsorbed to Cu(100). We explicitly treat nine modes corresponding to the motion of the C and O atoms and the Cu atom that is bonded to C. The potential used is a semi-empirical one developed by Tully and co-workers [J. C. Tully, M. Gomez, and M. Head-Gordon, J. Vac. Sci. Technol. A 11, 1914 (1993)], and is used fully, i.e., without recourse to multinomial expansion in displacement coordinates. We test the convergence of the results with respect to the number of modes coupled and find that the errors in the two-mode coupling representation vary from 0.6 to 6 cm−1 for the fundamentals but grow to 30 cm−1 for overtone and combination states. The errors in the three-mode representation of the potential are less than 0.2 cm−1 for the fundamentals and no larger than 2.5 cm−1 for high overtone/combination states with as much as 9 quanta of excitation. We calculate the thermally broadened spectra of the CO-stretch fundamental, the CO–Cu frustrated rotation and the CO–Cu frustrated translation over the temperature range 50–350 K. We compare the temperature dependence of the average frequency and standard deviation of these modes with experiment, and find semiquantitative agreement. © 1997 American Institute of Physics.
The vibrational self-consistent field (VSCF) and virtual configuration interaction (VCI) methods are directly combined with ab initio electronic structure calculations for evaluations of the potential energy at VSCF quadrature points. Referred to as direct VSCF and direct VCI, respectively, these methods have been applied to evaluations of anharmonic vibrational energy levels of H2O and H2CO at the second-order Møller–Plesset MP2/aug-cc-pVTZ and MP2/cc-pVTZ computational levels, respectively. The purpose of the present study is to develop a direct methodology for vibrational state calculations by examining the accuracy of the results, as well as their computational costs. In addition, the accuracy and applicability of two approximate potential energy surfaces (PES), a quartic force field (QFF), and the PES determined by the modified-Shepard interpolation method (Int-PES), are investigated via comparisons of calculated energy levels of vibrational states with those derived by the direct methods. The results are analyzed in terms of three considerations: (i) truncations of higher-order intercoordinate couplings in the PES; (ii) mode–mode coupling effects; (iii) approximations in ab initio electronic structure methods. In the direct VCI calculations, the average absolute deviations in fundamental frequencies relative to the experimental values are 9.3 cm−1(H2O) and 34.7 cm−1(H2CO). The corresponding values evaluated with approximate PESs relative to those derived by the direct method are 35.0 cm−1 (QFF) and 15.3 cm−1 (Int-PES) for H2O, and 6.3 cm−1 (QFF) and 10.3 cm−1 (Int-PES) for H2CO. © 2000 American Institute of Physics.
Vibrational energy levels, wave functions, and ir absorption intensities are computed for (H2O)n clusters with n=2, 3, 4, and 5. The calculations were carried out by the vibrational self‐consistent field (VSCF) approximation, with corrections for correlations between the modes by perturbation theory. This correlation corrected VSCF (CC‐VSCF) is analogous to the familiar Möller–Plesset method in electronic structure theory. Test calculations indicate that this method is of very good accuracy also for very anharmonic systems. While the method is of highest relative accuracy for the stiffest modes, it works very well also for the soft ones. Some of the main results are (1) the frequencies calculated are in good but incomplete agreement with experimental data available for some of the intramolecular mode excitations. The deviations are attributed to the inaccuracy of the coupling between intramolecular and intermolecular modes for the potential function used. (2) Insight is gained into the pattern of blue‐ or redshifts from the corresponding harmonic excitation energies for the various modes. (3) Anharmonic coupling between the modes dominates in general over the intrinsic anharmonicity of individual modes in determining the spectrum. (4) The anharmonic corrections to the frequencies of some intermolecular modes (shearing, torsional) are extremely large, and exceed 100% or more in many cases. (5) An approximation of quartic potential field in the normal mode displacement is tested for the clusters. It works well for the high and intermediate frequency modes, but is in error for very soft shearing and torsional modes. (6) The relative errors of the VSCF approximation are found to decrease with the cluster size. This is extremely encouraging for calculations of large clusters, since the VSCF level is computationally simple. © 1996 American Institute of Physics.
A redetermination of the quartic force field of the water molecule surface has been recently calculated by the method of Hoy, Mills, and Strey. A novel form of the bending coordinate has been introduced into this force field to ensure that the pure bending potential has the correct form at θ=π, and a few select force constant have been rerefined to the J=0 vibrational spectrum of H2O. The vibrational–rotational energy levels of the resulting surface have been determined by the variational method. Two particular sets have been examined: (i) low vibrational quanta, J≤9 and (ii) all levels below 12 000 cm−1 (relative to zero point), J≤2. The results are compared with the experimental levels in the literature.
A new implementation of the vibrational self-consistent field (VSCF) method is presented on the basis of a second quantization formulation. A so-called active terms algorithm is shown to be a significant improvement over a standard implementation reducing the computational effort by one order in the number of degrees of freedom. Various types of screening provide even further reductions in computational scaling and absolute CPU time. VSCF calculations on large polyaromatic hydrocarbon model systems are presented. Further, it is demonstrated that in cases where distant modes are not directly coupled in the Hamiltonian, down to linear scaling of the required CPU time with respect to the number of vibrational modes can be obtained. This is illustrated with calculations on simple model systems with up to 1 million degrees of freedom.
A family of multivariate representations is introduced to capture the input–output relationships of high‐dimensional physical
systems with many input variables. A systematic mapping procedure between the inputs and outputs is prescribed to reveal the
hierarchy of correlations amongst the input variables. It is argued that for most well‐defined physical systems, only relatively
low‐order correlations of the input variables are expected to have an impact upon the output. The high‐dimensional model representations
(HDMR) utilize this property to present an exact hierarchical representation of the physical system. At each new level of
HDMR, higher‐order correlated effects of the input variables are introduced. Tests on several systems indicate that the few
lowest‐order terms are often sufficient to represent the model in equivalent form to good accuracy. The input variables may
be either finite‐dimensional (i.e., a vector of parameters chosen from the Euclidean space Rn\mathcal{R}^n) or may be infinite‐dimensional as in the function space
\textCn [ 0,1 ]{\text{C}}^n \left[ {0,1} \right]. Each hierarchical level of HDMR is obtained by applying a suitable projection operator to the output function and each of
these levels are orthogonal to each other with respect to an appropriately defined inner product. A family of HDMRs may be
generated with each having distinct character by the use of different choices of projection operators. Two types of HDMRs
are illustrated in the paper: ANOVA‐HDMR is the same as the analysis of variance (ANOVA) decomposition used in statistics.
Another cut‐HDMR will be shown to be computationally more efficient than the ANOVA decomposition. Application of the HDMR
tools can dramatically reduce the computational effort needed in representing the input–output relationships of a physical
system. In addition, the hierarchy of identified correlation functions can provide valuable insight into the model structure.
The notion of a model in the paper also encompasses input–output relationships developed with laboratory experiments, and
the HDMR concepts are equally applicable in this domain. HDMRs can be classified as non‐regressive, non‐parametric learning
networks. Selected applications of the HDMR concept are presented along with a discussion of its general utility.
A compact and robust many-mode expansion of potential energy surfaces (PES) is presented for anharmonic vibrations of polyatomic
molecules, where the individual many-mode terms are approximated with various different resolutions, i.e., electronic structure
methods, basis sets, and functional forms. As functional forms, the following three representations have been explored: numerical
values on a grid, cubic spline interpolation, and a Taylor expansion. A useful index is proposed which rapidly identifies
important many-mode terms that warrant a high resolution. Applications to water and formaldehyde demonstrate that the present
scheme can increase the efficiency of the PES computation by a factor of up to 11 with the errors in anharmonic vibrational
frequencies being no worse than ~ 10cm−1.
We present an adaptive density-guided approach for the construction of Born–Oppenheimer potential energy surfaces (PES) in
rectilinear normal coordinates for use in vibrational structure calculations. The procedure uses one-mode densities from vibrational
structure calculations for a dynamic sampling of PESs. The implementation of the procedure is described and the accuracy and
versatility of the method is tested for a selection of model potentials, water, difluoromethane and pyrimidine. The test calculations
illustrate the advantage of local basis sets over harmonic oscillator basis sets in some important aspects of our procedure.
By substituting the standard mass-weighted normal coordinates with either Morse-like or Gauss-like coordinates, it is demonstrated that significant improvements can be made to the vibrational spectra of polyatomic molecules calculated variationally. Quartic force fields in the form of Taylor expansions are generated by density functional theory for water, formaldehyde and methane, and their vibrational spectra calculated by the perturbation normal coordinate code SPECTRO. These are then compared with three sets of spectra arising from the variational code MULTIMODE. Initial spectra are obtained using the identical Taylor expansion force fields. A subsequent set of spectra are then obtained for which the symmetric normal coordinates of the force fields are replaced by Morse-like coordinates and a final set of spectra are obtained for which the asymmetric normal coordinates of the force field are replaced by Gauss-like coordinates. The restriction is imposed that the complete set of derivatives to quartic are preserved under these coordinate transformations.
An automatic procedure for the generation of potential energy surfaces based on high level ab initio calculations is described. It allows us to determine the vibrational wave functions for molecules of up to ten atoms. Speedups in computer time of about four orders of magnitude in comparison to standard implementations were achieved. Effects due to introduced approximations--within the computation of the potential--on fundamental modes obtained from vibrational self-consistent field and vibrational configuration interaction calculations are discussed. Benchmark calculations are provided for formaldehyde and 1,2,5-oxadiazole (furazan).
We report full-dimensional,
ab initio
potential energy (PES) and dipole moment
surfaces (DMS) for water. The PES is a sum of one-, two- and three-body terms. The three-body potential is a fit, reported here, to roughly 30,000 intrinsic three-body energies obtained with second-order Møller–Plesset perturbation theory (MP2) and using the aug-cc-pVTZ basis set (avtz). The one- and two-body potentials are from an
ab initio
water dimer potential [Shank et al. , J. Chem. Phys.130, 144314 (2009)]. The predictive accuracy of the PES is demonstrated for the water trimer, tetramer, and hexamer by comparing the energies and harmonic frequencies obtained from the PES and new high level
ab initio calculations at the respective global minima. The DMS is constructed from one- and two-body dipole moments, based on fits to MP2/avtz dipole moments. It is shown to be very accurate for the hexamer by comparison with direct calculations of the hexamer dipole. To illustrate the anharmonic character of the PES one-mode calculations of the 18 monomer fundamentals of the hexamer are reported in normal coordinates.
We report three modifications to recent ab initio, full-dimensional potential energy surfaces (PESs) for the water dimer [X. Huang et al., J. Chem. Phys. 128, 034312 (2008)]. The first modification is a refit of ab initio electronic energies to produce an accurate dissociation energy D(e). The second modification adds replacing the water monomer component of the PES with a spectroscopically accurate one and the third modification produces a hybrid potential that goes smoothly in the asymptotic region to the flexible, Thole-type model potential, version 3 dimer potential (denoted TTM3-F) [G. S. Fanourgakis and S. S. Xantheas, J. Chem. Phys. 128, 074506 (2008)]. The rigorous D(0) for these PESs, obtained using diffusion Monte Carlo calculations of the dimer zero-point energy, and an accurate zero-point energy of the monomer, range from 12.5 to 13.2 kJ/mol (2.99-3.15 kcal/mol), with the latter being the suggested benchmark value. For TTM3-F D(0) equals 16.1 kJ/mol. Vibrational calculations of monomer fundamental energies using the code MULTIMODE are reported for these PESs and the TTM3-F PES and compared to experiment. A classical molecular dynamics simulation of the infrared spectra of the water dimer and deuterated water dimer at 300 K are also reported using the ab initio dipole moment surface reported previously [X. Huang, B. J. Braams, and J. M. Bowman, J. Phys. Chem. A 110, 445 (2006)].
We develop two approaches for growing a fitted potential energy surface (PES) by the interpolating moving least-squares (IMLS) technique using classical trajectories. We illustrate both approaches by calculating nitrous acid (HONO) cis-->trans isomerization trajectories under the control of ab initio forces from low-level HF/cc-pVDZ electronic structure calculations. In this illustrative example, as few as 300 ab initio energy/gradient calculations are required to converge the isomerization rate constant at a fixed energy to approximately 10%. Neither approach requires any preliminary electronic structure calculations or initial approximate representation of the PES (beyond information required for trajectory initial conditions). Hessians are not required. Both approaches rely on the fitting error estimation properties of IMLS fits. The first approach, called IMLS-accelerated direct dynamics, propagates individual trajectories directly with no preliminary exploratory trajectories. The PES is grown "on the fly" with the computation of new ab initio data only when a fitting error estimate exceeds a prescribed tight tolerance. The second approach, called dynamics-driven IMLS fitting, uses relatively inexpensive exploratory trajectories to both determine and fit the dynamically accessible configuration space. Once exploratory trajectories no longer find configurations with fitting error estimates higher than the designated accuracy, the IMLS fit is considered to be complete and usable in classical trajectory calculations or other applications.
We present a new methodology to perform fast correlation-corrected vibrational self-consistent field (CC-VSCF) calculations using ab initio potential energy points calculated on the fly. Our method is based on the replacement of all-electron basis sets with a pseudo-potential basis for heavy atoms, and on an efficient reduction of the number of pair-coupling elements used in the CC-VSCF procedure. The method is applied to several test systems: H2O, NH3, and CH4, where it proves to be efficient, providing a speedup factor of 2 compared to a standard CC-VSCF calculation. We also apply our technique to the simulation of the vibrational spectrum of ethane and show that very accurate results can be obtained with a substantial speedup for this system.
An automatic and general procedure for the calculation of geometrical derivatives of the energy and general property surfaces for molecular systems is developed and implemented. General expressions for an n-mode representation are derived, where the n-mode representation includes only the couplings between n or less degrees of freedom. The general expressions are specialized to derivative force fields and property surfaces, and a scheme for calculation of the numerical derivatives is implemented. The implementation is interfaced to electronic structure programs and may be used for both ground and excited electronic states. The implementation is done in the context of a vibrational structure program and can be used in combination with vibrational self-consistent field (VSCF), vibrational configuration interaction (VCI), vibrational Moller-Plesset, and vibrational coupled cluster calculations of anharmonic wave functions and calculation of vibrational averaged properties at the VSCF and VCI levels. Sample calculations are presented for fundamental vibrational energies and vibrationally averaged dipole moments and frequency dependent polarizabilities and hyperpolarizabilities of water and formaldehyde.
A highly accurate and efficient method for molecular global potential energy surface (PES) construction and fitting is demonstrated. An interpolating-moving-least-squares (IMLS)-based method is developed using low-density ab initio Hessian values to compute high-density PES parameters suitable for accurate and efficient PES representation. The method is automated and flexible so that a PES can be optimally generated for classical trajectories, spectroscopy, or other applications. Two important bottlenecks for fitting PESs are addressed. First, high accuracy is obtained using a minimal density of ab initio points, thus overcoming the bottleneck of ab initio point generation faced in applications of modified-Shepard-based methods. Second, high efficiency is also possible (suitable when a huge number of potential energy and gradient evaluations are required during a trajectory calculation). This overcomes the bottleneck in high-order IMLS-based methods, i.e., the high cost/accuracy ratio for potential energy evaluations. The result is a set of hybrid IMLS methods in which high-order IMLS is used with low-density ab initio Hessian data to compute a dense grid of points at which the energy, Hessian, or even high-order IMLS fitting parameters are stored. A series of hybrid methods is then possible as these data can be used for neural network fitting, modified-Shepard interpolation, or approximate IMLS. Results that are indicative of the accuracy, efficiency, and scalability are presented for one-dimensional model potentials as well as for three-dimensional (HCN) and six-dimensional (HOOH) molecular PESs.
The multiconfigurational time-dependent Hartree (MCTDH) approach uses optimized sets of time-dependent basis functions, called single-particle functions, to represent multidimensional wavefunctions and thereby facilitates efficient multidimensional quantum dynamics studies. The single-particle function bases are usually optimized for a single wavefunction. Here, an approach is studied which utilizes a common single-particle function basis to represent several wavefunctions simultaneously, i.e., the single-particle function basis is constructed to result in an optimized averaged description of a number of wavefunctions. The approach can favorably be used to obtain eigenstates of Hamiltonians or to represent thermal ensembles. Test calculations studying the vibrational states of CH(3) and the thermal rate constant of the H+CH(4)-->H(2)+CH(3) reaction are presented. It is found that the required size of the single-particle functions basis does not increase when the number of wavefunctions described simultaneously is increased. As a consequence, the new approach offers an increased efficiency, e.g., for MCTDH rate constant calculations.
An accurate and efficient method for automated molecular global potential energy surface (PES) construction and fitting is demonstrated. An interpolating moving least-squares (IMLS) method is developed with the flexibility to fit various ab initio data: (1) energies, (2) energies and gradients, or (3) energies, gradients, and Hessian data. The method is automated and flexible so that a PES can be optimally generated for trajectories, spectroscopy, or other applications. High efficiency is achieved by employing local IMLS in which fitting coefficients are stored at a limited number of expansion points, thus eliminating the need to perform weighted least-squares fits each time the potential is evaluated. An automatic point selection scheme based on the difference in two successive orders of IMLS fits is used to determine where new ab initio data need to be calculated for the most efficient fitting of the PES. A simple scan of the coordinate is shown to work well to identify these maxima in one dimension, but this search strategy scales poorly with dimension. We demonstrate the efficacy of using conjugate gradient minimizations on the difference surface to locate optimal data point placement in high dimensions. Results that are indicative of the accuracy, efficiency, and scalability are presented for a one-dimensional model potential (Morse) as well as for three-dimensional (HCN), six-dimensional (HOOH), and nine-dimensional (CH4) molecular PESs.
A fitting method of the sixth-order potential energy function is proposed, where ab initio potential energy data for the fitting are sampled in directions containing maximal anharmonic downward distortions detected by the scaled hypersphere search (SHS) method. This technique has been applied to H2O, HCHO, HCOOH, C2H4, CH3OH, CH3CHO, CH3NH2, B2H6, (H2O)2, and (H2O)3, where, without using the symmetry, 176, 904, 1432, 2992, 2520, 2760, 3608, 6232, 768, and 1456 times single-point energy calculations, respectively, were required for obtaining anharmonic terms. Experimental IR peak positions of not only fundamentals but also overtones and combinations in the excitation energy range of 1000-4000 cm(-1) could be reproduced very accurately by the post-vibrational self-consistent field theory employing potential functions obtained by the present SHS based polynomial fitting method.
A review is given on the multi-configuration time-dependent Hartree (MCTDH) method, which is an algorithm for propagating wavepackets. The formal derivation, numerical implementation, and performance of the method are detailed. As demonstrated by example applications, MCTDH may perform very eciently, especially when there are many (typically four to twelve, say) degrees of freedom. The largest system treated with MCTDH to date is the pyrazine molecule, where all 24 (!) vibrational modes were accounted for. The particular representation of the MCTDH wavefunction requires special techniques for generating an initial wavepacket and for analysing the propagated wavefunction. These techniques are discussed. The full efficiency of the MCTDH method is only realised if the Hamiltonian can be written as a sum of products of one-dimensional operators. The kinetic energy operator and many model potential functions already have this required structure. For other potential functions, we describe an e...