Diffusion Monte Carlo Calculations for Rare-earths: Hartree-Fock, Hybrid B3LYP, and Long-range Corrected LC-BLYP Functional

Our study aim is to make highlights on the failure or success of both popular B3LYP functional and long-range corrected LC-BLYP functional at different values of the range separation parameter in improving the ground state and excited states energies calculations for 4f-lanthanides and 5f-actinides. The most popular B3LYP functional fails to provide a balanced description of excitation energies for 4f-lanthanides. However, it gives reasonable results for the actinides with exception of Pu and Am atoms. The ground state energies of 5f-actinides are improved significantly at the critical value of the range separation parameter equals 0.35 a.u. while the optimal value for the excited states lies in the critical range of 0.35-0.4 a.u. However, our results show that LC-BLYP functional is not recommended for 4f-lanthanides which have strongly localized nature.


Introduction
Obtaining accurate results for rare-earths is still a challenging problem. The difficulties presented by these elements in quantum mechanical calculations arise from the large magnitude of the relativistic effect and the limitation in the electron correlation treatment. Quantum Monte Carlo (QMC) methods [1][2][3][4] are among the most accurate to predict relatively accurate properties of quantum systems. Besides its favorable scaling with system size, any arbitrarily complex wave function can be used because the integrals are evaluated numerically. Although there are many different QMC approaches, diffusion Monte Carlo (DMC) remains the most accurate one.
In recent years, there has been a concerned effort to improve the quality of QMC results in order to arrive to chemical accuracy. In fact, many papers [5][6][7] have employed the density functional theory (DFT) particularly the most successful functional Becke 3 parameter-Lee-Yang-Parr (B3LYP) as a starting point to produce orbitals to quantum Monte Carlo calculations and highly accurate results have been obtained. Nevertheless, the standard B3LYP functional contains only short-range HF exchange so the long-range asymptotic behavior still remains incorrect. The introduction of long-range HF exchange in DFT functional will correct the description of long-range exchange interaction. So the long-range corrected functionals are providing a considerable improvement with respect to the hybrid functionals. The general form of the long-range corrected where the traditional exchange-correlation functional xc E may be a pure or hybrid functional. The components labeled "LR" and "SR" are evaluated using the long-range and short-range Coulomb potentials, respectively, while HF C denotes the amount of HF exchange present in the original functional.
In this paper, we perform DMC calculations of the ground and excited states energies for some rare-earths by using both HF and DFT (B3LYP) orbitals to investigate the performance of each for f-elements. Indeed, we study the dependence of the long-range corrected LC-BLYP functional on the range separation parameter µ and examine its performance for rare-earths. The basic form of the wave function that we used is the Slater-Jastrow wave function which is considered the most common and simplest one. In the next section, we outline a brief description of the DMC method. The results are then presented and discussed. Finally, we give the conclusion of this work.

Computational Methods
Diffusion Monte Carlo method has been extensively described in the literatures [8][9][10] so we give here a brief description of it. The diffusion Monte Carlo (DMC) method is a stochastic projector method for solving the imaginary time many-body Schrödinger equation: where τ is the imaginary time,  this equation can be simulated with a random walk having diffusion, a draft, and a branching step and may be written in the integral form: Where the Green's function is a solution of the same equation (3) initial and can be interpreted as a probability of transition from a state R to R'. It is possible to use MC method to solve the integral in Eq. (4) but the difficulty is that the precise form of The evolution during the long time interval τ can be generated repeating a large number of short time stepsτ . In the limit , 0 → τ one can make use of the short time approximation for Green's function [11]: But due to the fermionic nature of electrons, the wave function must have positive and negative parts and this is opposite with the assumed nature of ψ which is a probability distribution. So the fixed-node approximation [12] had been used to deal with the fermionic antisymmetry which constrains the nodal surface of ψ to equal that of the antisymmetric trial wave function . T ψ In our calculations, the wave function was generated by using the quantum chemistry program GAMESS [13]. The Slater determinant was obtained from HF and DFT (B3LYP) methods in order to check the accuracy of each on this type of systems. We make use of CRENBL ECP [14] basis set for all elements except La atom where CRENBS ECP basis set is being used which proved to be successful for the calculations. The program package Qwalk [15]

Results and Discussion
We summarize our results for the ground and excited states for some rare earths using both HF and B3LYP orbitals in tables 1 and 2. The electronic excitation involves a promotion of an electron with a change of spin from 6s → 5d and 7s → 6d for lanthanides and actinides respectively. The excitation energies have been calculated directly from the difference between the excited and the ground state energy. In table 3 we list the calculated excitation energies along with the experimental values [16] for the sake of comparison. Unfortunately, no experimental data for Pm and Pu atoms are available.
It is clearly seen from the tables 1 and 2 that B3LYP performs better than HF for La, Ac, and Th, elements with no f-electrons. However, the situation is different when 4f and 5f subshells are being populated in lanthanides and actinides respectively.
At first our results show that for lanthanides having electrons in their 4f subshell, B3LYP performs worse than HF. The standard B3LYP functional produces unphysical results; the excited state energy is lower than the energy of ground state for many atoms. Indeed, self-consistent field (SCF) fails to converge on the first step in Gamess. As indicated in table 3 the calculated excitation energies employing B3LYP functional yield negative excitation energies for all 4f-lanthanides except for Ce and Pm atoms. These results confirm that the most popular B3LYP fails to calculate the excitation energy for 4f-lanthanides. In Fact, we believe that the poor performance of B3LYP functional in 4f-lanthanides is traced back to the strong localization of 4f-electrons. This failure is due to in part to the so called "self-interaction error" which means incomplete cancellation between the self-Coulomb term and approximate self-exchange contribution. On the other hand, the hybrid Universal Journal of Physics and Application 10(1): 5-10, 2016 7 B3LYP seems to be quite satisfactory for actinides especially in the case of Pa atom. However, for both Pu and Am atoms negative excitation energies have been obtained. The failure of standard B3LYP for Am atom is attributed to the well-known fact that starting with Am atom to the end of actinide series; the 5f-states are localized and resemble the 4f-states in lanthanides. So we predict that B3LYP functional is unable to deal with actinides at the end of the series. Otherwise, it is well known that in the hybrid DFT functional a part of the short-range HF exchange is mixed into the semi-local DFT exchange. The hybrid B3LYP functional contains 20% short range HF exchange. As has been described previously, the hybrid functional with only short-range HF exchange is insufficient to describe the ground and excited states of f-elements. In an attempt to study the influence of inclusion of long-range HF exchange on rare earths, we calculate the ground and the excited states energies by using the long-range corrected LC-BLYP functional (which combine long-range HF exchange with short-range BLYP exchange and using a standard full-range DFT correlation). To this aim, we study the performance of the latter functional at different values of the range separation parameter µ in comparison to pure BLYP functional for both lanthanides and actinides.
Note that Ce is the only one element exhibits 4f itinerant behavior so the observed improvement by using LC-BLYP is not surprising. On the other hand, despite Eu atom has strongly localized 4f-electrons, the use of LC-BLYP at the optimal value of µ enhances the results considerably which is a direct consequence of its half filled electronic configuration that acquires more stability to the atom.
It is also interesting to point out that although 4f-electrons are localized in Pr atom, a clearly improvement has been observed by applying the long range corrected scheme to BLYP functional. This behavior supports P. Lethuillier et al [17] observation who confirmed that the localization of praseodymium compounds is intermediate between those of cerium and neodymium.
The calculated results of actinides are listed in table 5. The ground state energies for U, Np, Pu, and Am atoms as a function of the range separation parameter µ ranging from 0.3 a.u. to 0.5 a.u. are indicated in Fig. 1. It should be noted that we also investigate the performance outside the range [0.3:0.5] a.u. but no significant improvement can be achieved.
Our calculations found that the addition of long-range HF exchange to pure BLYP functional does not alter the results significantly for Ac and Th atoms, actinide elements with no 5f-electrons, compared to the pure BLYP functional. In contrast, the presence of the long-range HF exchange clearly enhances the ground state energy for 5f-actinides, except Pa atom, at the value of 0.35 a. u. of the range separation parameter as indicated in Fig. 1. This value gives the lowest DMC ground state energy for all atoms indicated in the figure with the exception of the Pu atom where a comparable energy with the energy of µ =0.45 a.u. has been obtained.
Like Eu, the improvement gained for Am atom is related to its half filled electronic configuration. On the other hand, the excited state energies have been improved considerably at the value around the range 0.35 a.u.-0.4 a.u. which is slightly higher than the optimal value for the ground state.  Our recommended value for the range separated parameter whatsoever either in the ground or excited state is close to the value of 4 . 0 = µ a.u. suggested by Gerber et al. [18] for molecular systems. Otherwise, to some extent our value is comparable to the optimal value 45 . 0 = µ a.u. presented by Rohrdanz et al. [19] and Henderson et al. [20] for the ground state thermodynamic.
Finally, the results in this work support that LC-BLYP functional provides good results for 5f-actinides and some lanthanides having less localized character of 4f-electrons, however, this paper points out the shortcoming of the latter functional to deal with strongly localized systems. In fact, the inclusion of the long-range HF exchange corrects the asymptotic behavior of the exchange potential and removes the long-range self-interaction error which the conventional hybrid functionals is incapable to remove it. But from our point of view, the failure is mostly related to the LYP correlation which is not suited for strongly localized systems. So we intend to investigate the performance of applying the long-range corrected scheme to other exchange-correlation functionals not having the common LYP correlation in a future study.

Conclusions
By using the diffusion Monte Carlo (DMC) method, we investigate the performance of both HF and B3LYP orbitals for some rare earths. It was found that B3LYP functional works worse than Hartree-Fock for all 4f-lanthanides. Whereas, the same functional gives acceptable results for actinides with exception of Pu and Am atoms. We further test the performance of the long-range corrected LC-BLYP functional at different values of the range separation parameter µ . Unfortunately, the latter functional does not improve the calculations for most 4f-lanthanides which strongly localized character. Conversely, it enhances both the ground state energies of 5f-actinides at the value 35 . 0 = µ a.u. and the excited state energies around the range of 0.35-0.4 a.u.