Variational solution of congruent transformed Hamiltonian for many-electron systems using full configuration interaction calculation
The congruent transformation of the electronic Hamiltonian is developed to address the electron correlation problem in many-electron systems. The central strategy presented in this method is to perform transformation on the electronic Hamiltonian for approximate removal of the Coulomb singularity. The principle difference between the present method and the transcorrelated method of Handy and Boys is that the congruent transformation preserves the Hermitian property of the Hamiltonian. The congruent transformation is carried out using explicitly correlated functions and the optimum correlated transform Hamiltonian is obtained by performing a search over a set of transformation functions. The ansatz of the transformation functions are selected to facilitate analytical evaluation of all the resulting integrals. The ground state energy is obtained variationally by performing a full configuration interaction (FCI) calculation on the congruent transformed Hamiltonian. Computed results on well-studied benchmark systems show that for the identical basis functions, the energy from the congruent transformed Hamiltonian is significantly lower than the conventional FCI calculations. Since the number of configuration state functions in the FCI calculation increases rapidly with the size of the 1-particle basis set, the results indicate that the congruent transformed Hamiltonian provides a viable alternative to obtain FCI quality energy using a smaller underlying 1-particle basis set.
The form of the many-electron wavefunction in the proximity of the electron-electron and electron-nuclear coalescence point plays a critical role in accurate determination of the ground and excited state energies. Although, the precise structure of the many-electron wavefunction continues to be elusive, the form of the exact wavefunction at the coalescence point is well understood and is given by the Kato cusp condition. Kato (1957); Hammond et al. (1994); Lester (1997); Nightingale and Umrigar (1999) In the many-electron wavefunction, the electron-nuclear cusp condition can be incorporated by using Slater-type orbitals (STOs). For calculations involving Gaussian-type orbitals (GTOs), the one-electron basis can be improved iteratively by adding GTOs with increasing angular momentum quantum number.Klopper and Kutzelnigg (1986) The subject of convergence of single-particle basis has been analyzed extensively using both analytical and numerical techniques. Cuevas-Saavedra and Ayers (2012); Papajak and Truhlar (2012); Yanai and Shiozaki (2012) The electron-electron cusp has been the focus of intense research because of its direct relation to the electron correlation problem and accurate description of the Coulomb and Fermi hole. Frolov and Wardlaw (2011); Kurokawa et al. (2005); Nakatsuji and Nakashima (2009); Prendergast et al. (2001); Shiozaki and Werner (2010) However, unlike the electron-nuclear cusp, atom-centered basis functions are not ideal for accurate description of the many-electron wavefunction near the electron-electron cusp. Kong et al. (2012); Hättig et al. (2012); Clark et al. (2011) Indeed it has been shown that the slow convergence of a full configuration interaction (FCI) calculation with respect to the one-particle basis is related to the inadequate treatment of the electron-electron cusp. Hättig et al. (2012) The solution is to include explicit dependence in the form of the wavefunction, and there is a large assortment of quantum chemical methods that have incorporated this approach. For example, in the variational Monte Carlo (VMC) method, the Jastrow function is used for including explicit terms in the trial wavefunction. Nightingale and Umrigar (1999); Hammond et al. (1994) The form of the Jastrow is chosen to ensure that the electron-electron and electron-nuclear Kato cusp conditions are satisfied. The parameters in the Jastrow function are obtained by minimizing the linear combination of energy and its variance. Because of the complicated mathematical form of the Jastrow function it is not possible to evaluate the integral over the electronic coordinates analytically, and a stochastic numerical method is used for computation of the energy. Recently, Morales et al. performed highly accurate multi-determinant VMC calculations on water. Morales et al. (2012) A detailed review of various applications of Quantum Monte Carlo (QMC) methods in physics and chemistry can be found in Ref. Austin et al., 2012; Ceperley, 2010; Foulkes et al., 2001. Explicitly correlated methods have also been developed for post Hartree-Fock schemes such as perturbation theory (MP2-R12), coupled-cluster methods (CC-R12), and multireference CI schemes (R12-MRCI). These methods introduce the electron-hole interparticle distance directly into the calculation in order to increase the accuracy of the calculations. The field of explicitly correlated methods for electronic structure calculation has been reviewed and a detailed description of various methods can be found in Ref. Kong et al., 2012; Hättig et al., 2012; Ten-no, 2012. A common feature of the R12 and F12 methods discussed above is that they all involve analytical computation of the correlation function. Recently, Chinnamsetty and coworkers have presented an interesting study that compared and contrasted QMC with various F12 methods. Chinnamsetty et al. (2012)
A different strategy known as the transcorrelated method was developed by Handy and Boys in 1969. Boys and Handy (1969) The basic idea of the transcorrelated method is to remove the electron-electron Coulomb singularity by performing similarity transformation on the Hamiltonian using an explicitly correlated function. The method was later extended by Ten-no to treat the electron-electron cusp using Guassian geminal functions and was applied to chemical systems. Ten-no (2000); Yanai and Shiozaki (2012) The transcorrelated method has also been combined with other methods such as QMC Umezawa and Tsuneyuki (2003) and coupled-cluster theory Hino et al. (2002) and has been used to study electron correlation in periodic systems. Umezawa and Tsuneyuki (2004) One of the defining characteristics of this method is that the transcorrelated Hamiltonian is not Hermitian and therefore is not required to be bounded from below by the exact ground state energy. The correlation function can be obtained either by minimizing the energy variance of the transcorrelated Hamiltonian Handy (1971, 1972) or by requiring the correlation function to satisfy the electron-electron cusp condition.
The focus of the present work is to address the non-Hermitian property of the transcorrelated Hamiltonian by replacing the similarity transformation by congruent transformation. James et al. (1968); Datta (2004); Watkins (2002) By performing congruent transformation, we preserve the Hermitian property of the electronic Hamiltonian which allows us to use a standard electronic structure method such as configuration interaction method to minimize the total energy. The remainder of the paper describes the theoretical development and the implementation details of the method. The derivation of the congruent transformed Hamiltonian is presented in Sec. II. Details of performing FCI calculations using the congruent transformed Hamiltonian and interfacing it with existing FCI methods are presented in Sec. II.1 and II.2. Benchmark calculations using the congruent transformed Hamiltonian are presented in Sec. III. The analysis of the results and conclusions are presented in Sec. IV.
Ii Congruent transformed Hamiltonian
where is an explicitly correlated function which will be defined later. The expectation value of the CT Hamiltonian with respect to any trial wavefunction is given as
where . The above expression is mathematically equivalent to calculating the expectation value of the electronic Hamiltonian using a correlated wavefunction and is bounded from below by the exact ground state energy . The optimized energy associated with the CT Hamiltonian is obtained by performing a minimization with respect to the trial wavefunction and explicitly correlated function,
The optimization of the correlation function and the trial wavefunction is conducted in two steps. In the first step, the form of the trial function is kept fixed to a single Slater determinant and the parameters of the geminal functions are determined by minimizing the geminal parameters and the molecular orbitals. In the second step, the minimized geminal function is kept fixed and the trial wavefunction is minimized. The steps involved are described by the following equation
The optimization of the correlation function and the trial wavefunction are described in the following subsections.
ii.1 Optimization of the correlation function
The choice of the correlation function plays an important part in the implementation of the method for practical applications. In principle, a variety of correlated functions such as two and three-body Jastrow functions can be used. However, the matrix elements associated with these functions cannot be integrated analytically and one has to use numerical techniques such as the VMC method to calculate the integrals. In the present work, Gaussian-type geminal (GTG) functions were used for the correlated functions. The GTG functions were introduced by Boys Boys (1950, 1960) and Singer Singer (1960), and have been used extensively in explicitly correlated methods. Komornicki and King (2011); Korona et al. (1997); Bukowski et al. (1996, 1999); Chakraborty et al. (2008); Dahle et al. (2007) Slater determinants augmented with GTG functions have been used to study electron-electron and electron-proton systems. The integrals involving GTG functions with GTOs can be performed analytically and have been derived earlier. Persson and Taylor (1996, 1997); Boys (1950, 1960) The form of the correlated function used in the following calculations is defined as
where is number of electrons and is the number of Gaussian functions. The geminal coefficients in the GTG function are determined variationally. In the limit of , the energy becomes equal to the Hartree-Fock energy.
As a consequence, the HF energy is the upper bound to the geminal minimization process
The transformed Hamiltonian is expanded as a sum of 2-6 particle operators as shown below
where, the operators are defined by collecting all two, three, four, five and six particle operators obtained by expanding the summation in Eq. (10). Specifically,
The exact form of the operators have been derived earlier and are not duplicated here. Elward et al. (2012) It should be emphasized that the operators are defined so that they are completely symmetric with respect to all permutation of the indices
The operator is the permutation operator that belongs to the complete symmetric group . An important feature of this method is the availability of the analytical gradients of the total energy with respect to the geminal parameters. The gradients can be computed analytically and are given by the following expressions
The AO integrals involving the gradients of the GTG functions are performed analytically and are computed with other AO integrals.
ii.2 Optimization of the trial wavefunction
The optimization of the trial wavefunction is performed by performing a full configuration interaction (FCI) calculation on the congruent transformed Hamiltonian. The FCI wavefunction is constructed by performing all possible excitations from the reference wavefunction. Szabo and Ostlund (1989) This can be represented by the following expression,
where we have retained in the expression to emphasize that only a finite number of terms are evaluated. This point will be a subject of discussion later in the derivation. The occupied and virtual orbitals are represented by and , respectively, and the CI coefficients are represented by and are obtained variationally by minimizing the total energy. The construction of the full set of excitations and the determination of the CI coefficients are the two principle computational challenges associated with the FCI method. For very small molecules, the CI matrix can be explicitly constructed and diagonalized, however, this simple approach becomes prohibitively expensive as the system size increases. Currently, there are various computational techniques for efficient calculation of the expansion coefficients. Bak et al. (2001); Blundell and Joshi (2010); Boguslawski et al. (2011); Corni et al. (2003); Prendergast et al. (2001); Rontani et al. (2006) The calculation requires matrix elements involving the operators which are derived below.
The matrix elements involving the 2-particle operators are evaluated as
The matrix elements involving the 3-particle operators are evaluated as
The matrix elements involving the 4-particle operators are evaluated as
The matrix elements involving the 5-particle operators are evaluated as
The matrix elements involving the 6-particle operators are evaluated as
The computation of matrix elements in the above expression requires atomic orbital integrals involving the GTG functions. One of the advantages of using the GTG functions is that all the AO integrals needed for the CT Hamiltonian calculation can be computed analytically. Boys and Singer have derived the integrals involving GTG functions with s-type GTOs. Persson and Taylor have extended the method for higher angular momentum by using the Hermite Gaussian expansion approach. Persson and Taylor (1997) Recently, Hofener and coworkers have also derived the geminal integrals by extending the Obara-Saika techniques for calculating the GTG integrals. HÃ¶fener et al. (2009)
The solution for the CI coefficients requires diagonalization of the CI Hamiltonian matrix. However, the lowest eigenvalue and eigenfunction can be obtained without explicit construction and storage of the CI matrix. There are various efficient methods such as the Davidson diagonalization to perform this task. Davidson (1975) Recently, Alavi et al. have developed the FCIQMC method which allows very efficient evaluation of the FCI wavefunction. Booth et al. (2009); Booth and Alavi (2010); Booth et al. (2011); Cleland et al. (2011); Shepherd et al. (2012)
In the present calculation, the FCI eigenvector was obtained by performing the Nesbet update scheme and was selected because of its ease of implementation. Nesbet (1965) In the Nesbet method, a expansion coefficient is updated by
where the update is calculated as
The energy is updated at each step using
The FCI energy can be recovered from the CT calculation by setting
From the above relationship, we expect that the CTH energy calculated with will be lower than the FCI results. In the following section, we perform CTH calculations on well-studied two-electron systems and compare calculated energies with reported benchmark values.
Iii Calculations and results of benchmark systems
The Hooke’s atom is one of the few correlated two-electron systems for which the Schrödinger equation can be solved analytically. This feature has made it a testing ground for a wide variety of methods. Joubert (2012); Nakatsuji (2005); Yang et al. (2011); Liang et al. (2011) The Hooke’s atom consists of two electrons in a parabolic potential. The Hamiltonian of that system can be written as
where, all the quantities are expressed in the atomic units. The interaction between an electron and the nucleus is described with the harmonic potential. For , the Schrödinger equation can be solved exactly and the ground state energy is equal to 2.0 Hartrees. Matito et al. (2010) The Hooke’s atom provides an ideal ground for testing the CTH method. The CTH calculations were performed using the 6-311G basis and the geminal parameters were obtained variationally from the solution of Eq. (4). The energy was converged with respect to number of geminal parameters , and the results are presented in Fig. 1.
It is seen that the energy was converged after addition of four geminal parameters and the optimized geminal parameters are listed in Table 1.
Comparing the energy with the exact result of 2.0 Hartrees, it is seen that the is slightly higher by 0.770 mHartrees (or 0.483 kcal/mol). The optimized Slater determinant obtained in the previous step is used as the reference wavefunction for the CTH calculations and the results are summarized in Fig. 1. For , the CTH energy is identical to the FCI energy. However, inclusion of additional geminal terms makes the CTH energy lower than the energy from the FCI calculation. It is seen that the CTH energy is in good agreement with the exact analytical results and is higher by 0.000296 Hartrees, these results are provided in Table 2.
The CTH calculations were also carried out for the helium atom and the results are presented in Fig. 2.
The calculations were performed using different basis functions, and the results were compared with HF and FCI values. It is seen that for small basis sets, the energy is lower than the FCI energy. We expect this because of the inclusion of the optimized geminal terms. The key result from Fig. 2 is that for small basis sets, the CTH method provides a substantial lowering of energy with respect to the corresponding FCI values. The CTH calculations with respect to a small basis provides a wavefunction that is comparable to the FCI wavefunction at much larger basis functions. Since the cost of the FCI expansion increases sharply with the size of the underlying 1-particle basis, the CTH method provides an appealing alternative for obtaining accurate results when an FCI calculation is prohibitively expensive. The dependence of the CTH energies on the number of geminal parameters is shown in Fig. 3
and the optimized geminal parameters for the helium atom are listed in Table 3.
Iv Discussion and conclusions
The first geminal parameter is always set to and and is never optimized during the calculations. When all the other geminal parameters are set to zero, these values of and represent the limit. Geminal parameters from and are optimized to obtain as described in Eq. (4). This procedure ensures that the optimized energy is always bounded from above by the HF energy. Figures 1 and 2 show the effect of inclusion of additional geminal parameters and it is seen that the second geminal parameter lowers the energy significantly. This is an important result and clearly indicates the importance of the geminal function in construction of the congruent transformed Hamiltonian. The set of was optimized without any constraint and it is seen from Tables 1 and 3 that the overal geminal parameter is negative. This is an expected result and is in agreement with previous work on explicitly correlated methods. HÃ¶fener et al. (2009); Shiozaki et al. (2011); Ten-no and Noga (2012) The negative values of geminal parameters indicate the role of the geminal function in providing a better description of the Coulomb hole.
The analytical forms of the GTG functions are inherently approximate and are not capable of describing the cusp correctly because their first derivative vanishes in the limit of
To assess the quality of the CTH energy, it is important to estimate how much of an error this feature introduces in the calculated energy. For the Hooke’s atom this can be done in a a straightforward manner since the analytical solution of the Schrödinger equation is known. From Table 2, it is seen that the CTH energy is close to the exact ground state energy and is higher by 0.296 mHartrees or 0.186 kcal/mol. This difference between the CTH and the exact energy represents the upper bound in the error that one can expect for this system by approximating the cusp with GTG functions. For the helium atom, the situation is less straightforward because we do not have access to the exact solution. Instead, we compared the CTH energies with other high-level methods from previous studies Hylleraas (1929); Kinoshita (1959, 1957); Nakashima and Nakatsuji (2008, 2007); Rosendorff and Schlaile (1989); May et al. (2005); Rosendorff and Schlaile (1993); Morgan (1978) that include the exact cusp condition in the wavefunction. In order to achieve the best CTH energy, the calculation was performed with an aug-cc-pVTZ basis set and geminal parameters were optimized with respect to the aug-cc-pVTZ basis. Comparing the CTH method with the highly accurate ICI method by Nakatsuji Nakashima and Nakatsuji (2008, 2007), it is seen that CTH energy is higher than the ICI energy by 0.429 kcal/mol. The comparison of the CTH calculation to the ICI method and other highly accurate results can be seen in Table 4.
|-2.900233||FCI||III Johnson (2011)|
|-2.903724||free ICI||Nakashima and Nakatsuji (2008, 2007)|
The impact of electron-electron cusp on ground state energy was investigated in detail by Prendergast Prendergast et al. (2001) and coworker using CI and QMC methods. Their study concluded that one can still expect to get mHartree level of accuracy even in situations where the exact cusp condition is not satisfied. Our study using GTG functions also confirms this observation. The use of GTG functions in the CTH method, represents a trade-off between the implementation of exact cusp condition and analytical expression for computing the Gaussian-type geminal integrals.
In conclusion, the congruent transformation of the electronic Hamiltonian using Gaussian type geminal function is presented as a general method for calculating accurate ground state energy. The form of the congruent transformed Hamiltonian can be systematically improved by using the geminal function. It was found that a small number of geminal functions are needed to converge the energy. Futhermore, addition of just one geminal parameter results in a substantial improvement in the accuracy of the wave function. For a given finite basis set the CTH energy was found to be lower that the FCI calculation on untransformed Hamiltonian. The results indicate that the congruent transformed Hamiltonian provides a viable alternative for obtaining FCI quality energy using a smaller underlying 1-particle basis set.
Acknowledgements.We gratefully acknowledge the support from Syracuse University, Graz University of Technology, NSF REU program Grant No. CHE-0850756, and NSF iREU program Grant No. CHE-0755383 for this work.
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