Correlated two-electron momentum properties for helium to neon atoms

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1 JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER MARCH 1999 Correlated two-electron momentum properties for helium to neon atoms A. Sarsa, F. J. Gálvez, a) and E. Buendía Departamento de Física Moderna, Facultad de Ciencias, Universidad de Granada, E Granada, Spain Received 29 September 1998; accepted 23 December 1998 Two-electron properties in momentum space for the atoms helium to neon have been calculated starting from explicitly correlated wave functions. The different integrals involved in the calculation have been evaluated by using the Monte Carlo algorithm. In particular, the spherically averaged interelectronic momentum distribution, (2) (p 12 ), its radial moments p n 12, with n 2to 3, the expectation value p 1 p 2, and both the electron electron coalescence, (2) (0), and counterbalance, (2) (0), densities have been calculated. A systematic study of the electronic correlation has been performed by comparing the correlated results with the corresponding Hartree Fock ones. Finally an analysis of the structure of the interelectronic momentum distribution in terms of its parallel and antiparallel components has been carried out American Institute of Physics. S I. INTRODUCTION Position and momentum space properties provide complementary information to describe the structure of electronic systems. While the former have been extensively studied for atoms by using both Hartree Fock and correlated wave functions, in momentum space the information is more scarce due mainly to the technical difficulties which appear in their calculation. An important quantity in momentum space is the single particle density, (1) (p), because some of its radial expectation values, p n, have a special physical significance and also because one can obtain from it the atomic Compton profile, within the impulse approximation, which is an experimentally measurable quantity. 1 Different calculations of the one body momentum density distribution have been carried out both within the Hartree Fock framework, 2,3 leading to extensive tabulations of Hartree Fock atomic momentum space properties, 4 6 and with the use of correlated wave functions, mainly within a configuration interaction scheme, 7 10 although explicitly correlated wave functions have been also used in studying two electron systems Two-electron momentum density can be analyzed in terms of the interelectronic momentum, (2) (p 12 ), and the center of mass, (2) (P), densities, 14 defined as the expectation values 2 p 12 and N p 1 1,...,p N N p 1 1,...,p N N respectively. Here i is the spin coordinate and (p 1 1,...,p N N ) stands for the momentum space wave function. These quantities are the probability density function for a pair of electrons having a relative momentum p 12 or a center of mass momentum P, respectively. Their spherical averages are denoted by (2) (p 12 ) and (2) (P), respectively. These densities have been much less studied than their counterpart in position space due to the numerical difficulties that appear in the calculation of the momentum wave function. Let us remark the work of K.E. Banyard and coworkers for two 15,16 and three 17 electron atoms within a configuration interaction scheme and, more recently, the calculation of the interelectronic moments p n 12, with n 2 to 3, 18 and the value of the electron electron coalescence, (2) (0), and counterbalance, (2) (0), densities in momentum space, 19 performed by calculating explicitly the momentum space wave function from the numerical solution of the Hartree Fock equations. 20 In this work we use the Monte Carlo MC method to deal with the integrals involved in the calculation of these two-electron momentum space properties. This technique allows one to evaluate the expectation value of any operator between wave functions of any type, becoming a powerful tool in quantum chemistry calculations. 21 The main aim of this work is, by using the MC method and working with explicitly correlated wave functions, to calculate several twoelectron momentum properties for the atoms helium to neon. In particular we shall calculate the electron electron coalescence, (2) (0), and the electron electron counterbalance, (2) (0), that represent the probability densities for any pair of electrons to have the same and opposite momenta, respeca Electronic mail: galvez@goliat.ugr.es k j 1 p 12 p k p j 1 2 P N p 1 1,...,p N N p 1 1,...,p N N k j 1 P p k p j / /99/110(12)/5721/7/$ American Institute of Physics

2 5722 J. Chem. Phys., Vol. 110, No. 12, 22 March 1999 Sarsa, Gálvez, and Buendía tively. We have also calculated the spherically averaged interelectronic momentum distribution, (2) (p 12 ), and the expectation values p n 12, with n 2to 3. Finally we shall also obtain the expectation values p 2 12 and p 1 p 2 in terms of the expectation values i, j ı j 2 and i, j i j, respectively. All these quantities are compared with their respective Hartree Fock values in order to study the electronic correlation. Finally we have also studied the parallel and antiparallel components of (2) (p 12 ), i.e. the interelectronic momentum density due to pair of electrons with the same and opposite spin, respectively. The structure of this work is as follows. In Sec. II we discuss the computational method used to calculate all the quantities studied in this work. In Sec. III we describe the functional form of the wave functions we have worked with, giving their main properties. In Sec. IV we show all the correlated results obtained in the present work. Finally, in Sec. V we give the main conclusions of the present study. Atomic units are used throughout this paper. II. MONTE CARLO METHOD AND MOMENTUM DISTRIBUTIONS As we have mentioned above the calculation of the interelectronic momentum density, (2) (p 12 ), starting from Eq. 1 is not an easy task due to the difficulties in calculating the momentum wave function,, even more if the position wave function depends explicitly on the interelectronic coordinates r ij, i.e., for explicitly correlated wave functions. However, it is possible to express (2) (p 12 ) in terms of the position wave function,, as 2 p 12 N d r dr 1 2 r,r 1 1, 2 eıp 12 r 1 r where (2) (p 12 ) is the contribution to (2) (p 12 ) from all the electron pairs with spin, d r means dr 1...,dr N, the sum runs over 1, 1, 2 and 2, N is the number of electron pairs with spin, and 2 r,r 1 1, 2 * r 1 1,...,r N N r 1 1,r 2 r 1 r 1 2,r 3 3,...,r N N 4 Now the Monte Carlo algorithm can be used to calculate (2) (p 12 ). In doing so it is convenient to use the identity 1 dr 8 1 e r d r dr 1 e r 1 * r 1 1,...,r N N r 1 1,...,r N N, which allows one to write Eq. 3 in the form f :g; Vd f f ; V d g g;, 6 where ( ) stands for the distribution function that, as a first option for almost all the atoms here studied, has been chosen as 2 ( r,r 1 1, 2 ). The results obtained with it have been labeled as MC 1. Once obtained the interelectronic momentum density one can evaluate the expectation values p n 12 defined as p n 12 d p 12 p n 12 2 p 12. However, the Monte Carlo algorithm applied to determine the spherically averaged momentum distribution (2) (p 12 ) is able to calculate it adequately up to a given value of the interelectronic momentum from where it starts to oscillate around the exact density for example, for neon within the Hartree Fock framework and working with 10 8 movements for each one of the electrons the oscillations appear for p a.u.. Then to integrate from this point to we have used the asymptotic behavior of the momentum distribution given by 2 p 12 D 8 p D 10, p 12 that has been shown as correct for hydrogenlike wave functions 18 and that we conjecture holds for the different wave functions used in the present work. 22 This asymptotic behavior is similar to the one found for the single particle momentum density. 23 The coefficients D j, which have alternate signs, are calculated by fitting the asymptotic expression 8 to the corresponding Monte Carlo distribution. This has been done by minimizing the difference between the calculated and the exact values of both the norm and the expectation value p The method used here to determine the different momentum properties has been tested within the Hartree Fock framework. In Table I we show, for the atoms helium to boron, the values of (2) (0) and the expectation values p n 12, n 2, 1,1,2, and 3 obtained with the Monte Carlo quadrature MC 1 row by using the wave functions of Clementi and Roetti 24 as compared with the values obtained from the numerical Hartree Fock solution. 18,19 For helium and lithium the values of the density at the origin and the moments of negative order are quite well reproduced with the Monte Carlo method. However, for the beryllium atom the momentum distribution is underestimated for low values of the interelectronic momentum p 12 as can be noticed from the values of (2) (0) and the moments of negative order. This bad behavior also holds for boron and carbon. To solve this problem without increasing the computer time we have used in the Monte Carlo sampling another distribution function that increases the statistics in the low p 12 region. The alternative distribution function used in the present work for these three atoms is the square of the modulus of the wave function times exp r 1, i.e., the denominator in Eq. 3. We have checked that this distribution function amends the values of (2) (p 12 ) for short values of the argument although the momentum distribution obtained with it starts to oscillate for smaller p 12 values than in the previous case. It is possible, however, to use the correct behavior of each one of the 7

3 J. Chem. Phys., Vol. 110, No. 12, 22 March 1999 Sarsa, Gálvez, and Buendía 5723 TABLE I. Values of some interelectronic momentum properties calculated within the Hartree Fock framework for the atoms helium to boron obtained with the Monte Carlo algorithm as compared with the numerical Hartree-Fock calculation of Refs. 18 and 19. The rows MC 1 and MC 2 correspond to two different distribution functions in the Monte Carlo calculation. In parentheses we give the statistical error in the last digit. See text for more details. (2) (0) p 2 12 p 1 12 p 1 12 p 2 12 p 3 12 He ( 1 S) MC a b b b b 21.9 b Li ( 2 S) MC MC a b b b b b Be ( 1 S) MC MC a b b b b 600 b B( 2 P) MC MC a b b b b 1700 b a Numerical Hartree Fock calculation of Koga and Matsuyama Ref. 19. b Numerical Hartree Fock calculation of Koga and Matsuyama Ref. 18. momentum densities obtained with the use of the corresponding distribution function by connecting them at an intermediate point. The results thus obtained for lithium, beryllium, and boron are also given in Table II MC 2 row. Itis apparent the improvement in (2) (0) and in the different momentum expectation values, mainly in p 2 12 and p The atoms nitrogen to neon show the same behavior as helium and lithium. III. CORRELATED WAVE FUNCTION The structure of the correlated wave function,, used in this work is the product of a symmetric correlation factor, F, which includes the dynamic correlation among the electrons times a model wave function,, that provides the correct properties of the exact wave function such as the spin and the angular momentum of the atom, and is antisymmetric in the electronic coordinates. F. 9 For the correlation factor we use the form of Boys and Handy 25 U F e ij i j, 10 being and N c U ij k 1 r i m c k n k n k m k o k k r i r j r i r j r ij 11 r i, r ij r ij r i 1 r ij We have used the values of m k, n k, and o k proposed by Schmidt and Moskowitz. 26 This part of the wave function can include 7, 9, or 17 variational parameters. The model wave function,, has been chosen in some different ways. In the first one we have taken the Hartree Fock solution tabulated by Clementi and Roetti 24 and the total wave function has been denoted by 7, 9, and 17. For beryllium, boron, and carbon we have also considered the 2s-2p near degeneracy effect 27 by using the multideterminant wave function 1 2, 13 where 1 and 2 are the Hartree Fock solutions corresponding to the configurations 1s 2 2s 2 2p k and 1s 2 2p k 2, respectively, and k 0,1,2 for beryllium, boron, and carbon, respectively. Here is a new variational parameter and the total wave function has been denoted by n,1. Finally, for the beryllium atom, we have also considered the variation of in the minimization process. The wave function thus obtained has been denoted by 10,1. The ground state energy and different properties provided by these wave functions can be found in Refs. 26, and The best energy for each atom is obtained with 17 for helium and for nitrogen to neon although for the latter the wave function 9 provides practically the same energy as 17 ), with 17 for lithium, with 17,1 and 10,1 for beryllium and, finally, with 17,1 for boron and carbon. However, a detailed study of one- and two-electron densities in position space leads to the conclusions that 17 for lithium and 17,1 for beryllium provide densities that are very extended in space and that although 7 only recovers about 90% of the correlation energy for the lithium atom it describes adequately both the single particle and the electron pair density. 22,31 Besides 9 and 17, for the neon atom provide a quite similar single particle density, even a bit better for the former for large values of the electron coordinate. Thus the correlated wave functions we have worked with in the present work are 17 for helium and for nitrogen

4 5724 J. Chem. Phys., Vol. 110, No. 12, 22 March 1999 Sarsa, Gálvez, and Buendía TABLE II. Correlated interelectronic momentum properties for the atoms helium to neon compared with the numerical Hartree Fock HF ones and with others obtained with an configuration interaction CI wave function for helium and lithium. In parentheses we give the statistical error in the last digit. (2) (0) p 2 12 p 1 12 p 12 p 2 12 p 3 12 He ( 1 S) a a b b b 21.9 b c Li ( 2 S) a b b b b b d Be ( 1 S) a b b b b 600 b 10, B( 2 P) a b b b b 1700 b 17, C( 3 P) a b b b b 3750 b 17, N( 4 S) a b b b b 7350 b O( 3 P) a b b b 1059 b b F( 2 P) a b b b 1610 b b Ne ( 1 S) a b b b 2344 b b a Numerical Hartree Fock calculation of Koga and Matsuyama Ref. 19. b Numerical Hartree Fock calculation of Koga and Matsuyama Ref. 18. c CI calculation of Banyard and Reed Ref. 15. d CI calculation of Banyard and Youngman Ref. 17. to fluorine, 7 for lithium, 10,1 for beryllium, 17,1 for boron and carbon, and 9 for neon. IV. CORRELATED TWO-ELECTRON MOMENTUM PROPERTIES Correlated two-electron momentum density data do no exist in the literature to the best of our knowledge, except for very light atoms. In Table II we give the correlated values of (2) (0) and p n 12, with n 2 to 3, as compared with the noncorrelated ones of Refs. 18 and 19 and with others correlated values for helium and lithium. 15,17 As we can see the correlated value of (2) (0) is smaller than the noncorrelated one for helium and neon, it does not modify appreciably its value in carbon and is bigger in all the other atoms. Thus, (2) (p 12 ) does not show a structure that allows one to conclude about the existence of a Coulomb hole in momentum space such as the one that holds in position space. The correlated moments p n 12 shown in Table II are smaller than the corresponding Hartree Fock ones for helium and lithium. For beryllium, boron, and oxygen, the correlated moments of negative order are larger than the corresponding Hartree Fock ones while the moments of positive order are smaller. On the other hand, the correlated interelectronic moments of negative order of nitrogen, fluorine, and neon are smaller than the Hartree Fock ones while the moments of positive order are larger. For the carbon atom the correlated values practically coincide with the noncorrelated ones. In Figs. 1 and 2 we plot the correlated (2) (p 12 ) distribution and its parallel, p (2), and antiparallel, a (2), components for all the atoms here studied. The most important contribution to (2) (p 12 ) arises from those electrons with antiparallel spin. However, the parallel component is responsible for the maximum of (2) (p 12 ) for all the atoms other than helium for which it is zero, beryllium, boron, and carbon. In these last atoms the parallel component is less important than in the others, mainly in beryllium where it is practically negligible. In Table III we summarize some other two electron momentum properties obtained in the present work as compared with the Hartree Fock ones. First we show the value of the correlated electron electron counterbalance density, (2) (0), obtained from d r dr 1 2 r,r 1 1, N , 14

5 J. Chem. Phys., Vol. 110, No. 12, 22 March 1999 Sarsa, Gálvez, and Buendía 5725 FIG. 1. Parallel, p (2), antiparallel, a (2), and total interelectronic momentum density, (2) (p 12 ), for the atoms helium to carbon. where 2 r,r 1 1, 2 * r 1 1,...,r N N r 1 1,r 2 r 1 r 1 2,r 3 3,...,r N N, 15 and the other quantities are defined as in Eq. 3. Some comments about this quantity are the following: i As one can see this quantity is quite sensitive to the electronic correlations. Besides, in all the atoms considered here, the correlated (2) (0) value is smaller than the noncorrelated one, the beryllium atom being the one for which this effect is more accused. This decrease is less important for heavier atoms. ii Recently 32 has been shown that the Hartree Fock electron electron coalescence, (2) (0), and counterbalance, (2) (0), fulfill the inequality , 16 the equality holds for helium, lithium, and beryllium. We have numerically checked that, in general, this inequality is not fulfilled for correlated wave functions, because the effect of the electronic correlation is to decrease the value of the counterbalance density, (2) (0), while does not modify appreciably the value of (2) (0). iii Finally T. Koga 33 has introduced the concept of electron electron counterbalance hole in Hartree Fock theory by showing that two electrons with the same spin and the same spatial inversion symmetry do no contribute to the value of (2) (0). We have numerically checked that for explicitly correlated wave functions the contribution of a such pair of electrons is negligible with respect to the total value of the counterbalance density, and can be considered as zero, within the statistical error. We also show in Table III the value of i, j ı j 2, i.e., the expectation value p 2 12, obtained from both the Hartree Fock and the correlated wave functions with the Monte Carlo algorithm. These values have been used to determine the coefficients D j that determine the asymptotic behavior of (2) (p 12 ) given by Eq. 8. Finally, in this table, we also give the value of i, j i j which is an important quantity in the determination of both the specific mass shift correction to the total ground state energy, and the transition isotope shift, 34 and, as it is known, 35 is also a sensitive mea-

6 5726 J. Chem. Phys., Vol. 110, No. 12, 22 March 1999 Sarsa, Gálvez, and Buendía FIG. 2. Parallel, p (2), antiparallel, a (2), and total interelectronic momentum density, (2) (p 12 ), for the atoms nitrogen to neon. sure of the inadequate description of electron correlation. This quantity has a zero value for helium, lithium, and beryllium and is positive for all the rest of atoms within the Hartree Fock framework. The effect of the electronic TABLE III. Correlated values for (2) (0) and some differential operators as compared with Hartree Fock ones. In parentheses we give the statistical error in the last digit. (2) (0) i,j ı j 2 i,j ı j He ( 1 S) HF a Li ( 2 S) HF a Be ( 1 S) HF a , B( 2 P) HF a , C( 3 P) HF a , N( 4 S) HF a O( 3 P) HF a F( 2 P) HF a Ne ( 1 S) HF a a Numerical Hartree Fock calculation of Koga and Matsuyama Ref. 19. correlation is to decrease its value with respect to the Hartree Fock one in such a way that the correlated values are negative for the atoms helium to boron and positive for carbon to neon. Let us mention here that our correlated results coincide, within the statistical error, with those of Ref. 29 for helium, nitrogen, oxygen, and fluorine, i.e., for those atoms which are described with the 17 wave function. For the other atoms we have used a different wave function. V. CONCLUSIONS By using the Monte Carlo algorithm, we have calculated several two electron properties in momentum space for the atoms helium to neon starting from the explicitly correlated wave functions of Schmidt and Moskowitz 26 and a generalization of them to include the nondynamic effect due to the 2s 2p near degeneracy in the atoms beryllium, boron, and carbon. We have also modified the central part of the wave function in the minimization process for the beryllium atom. An analysis of the influence of the electronic correlation on different two-electron properties has been done. We have found that the electronic correlation does not present a systematic effect on the interelectronic momentum distribution as it happens for its counterpart in position space. The quantities for which the effects of the electronic correlation are more important, for all the atoms here studied, are (2) (0) and i, j i j. We have found that the electronic correlation decreases their values with respect to the Hartree Fock ones in all the systems here considered. Finally we have analyzed the structure of the interelectronic momentum density, (2) (p 12 ), in terms of its components parallel and antiparallel.

7 J. Chem. Phys., Vol. 110, No. 12, 22 March 1999 Sarsa, Gálvez, and Buendía 5727 ACKNOWLEDGMENTS This work has been partially supported by the Spanish Dirección General de Investigación Científica y Técnica DGICYT under contract PB A and by the Junta de Andalucía. 1 B. G. Williams, Compton Scattering McGraw-Hill, New York, A. M. Simas, W. M. Westgate, and V. H. Smith, Jr., J. Chem. Phys. 80, W. M. Westgate, A. M. Simas, and V. H. Smith, Jr., J. Chem. Phys. 83, A. J. Thakkar, A. L. Wonfor, and W. A. Pedersen, J. Chem. Phys. 87, J. M. G. Vega and B. Miguel, At. Data Nucl. Data Tables 54, ; 58, ; 60, T. Koga and A. J. Thakkar, J. Phys. B 29, R. O. Esquivel, A. N. Tripahi, R. P. Sagar, and V. H. Smith, Jr., J. Phys. B 25, A. N. Tripahi, R. P. Sagar, R. O. Esquivel, and V. H. Smith, Jr., Phys. Rev. A 45, A. N. Tripahi, V. H. Smith, Jr., R. P. Sagar, and R. O. Esquivel, Phys. Rev. A 54, H. Meyer, T. Müller, and A. Schweig, J. Mol. Struct.: THEOCHEM 360, R. Benesch, J. Phys. B 9, P. E. Regier and A. J. Thakkar, J. Phys. B 18, F. Arias de Saavedra, E. Buendía, and F. J. Gálvez, Z. Phys. D 38, A. J. Thakkar, in Density Matrices and Density Functionals, edited by R. Erdahl and V.H. Smith, Jr. Reidel, Dordrecht, 1987, p K. E. Banyard and J. C. Moore, J. Phys. B 10, ; K. E. Banyard and C. E. Reed, ibid. 11, P. K. Youngman and K. E. Banyard, J. Phys. B 20, K. E. Banyard and P. K. Youngman, J. Phys. B 20, ; K.E. Banyard, K. H. Al-Bayati, and P. K. Youngman, J. Phys. B 21, T. Koga and H. Matsuyama, J. Chem. Phys. 107, T. Koga and H. Matsuyama, J. Phys. B 30, C. Froese Fisher, Comput. Phys. Commun. 4, B. L. Hammond, W. A. Lester, Jr., and P. J. Reynolds, Monte Carlo Methods in ab initio Quantum Chemistry World Scientific, Singapore, A. Sarsa, Ph. D. University of Granada J. C. Kimball, J. Phys. A 8, E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14, S. F. Boys and N. C. Handy, Proc. R. Soc. London, Ser. A 310, K. E. Schmidt and J. W. Moskowitz, J. Chem. Phys. 93, J. Linderberg and H. Shull, J. Mol. Spectrosc. 5, J. W. Moskowitz and K. E. Schmidt, J. Chem. Phys. 97, S. A. Alexander and R. L. Coldwell, J. Chem. Phys. 103, A. Sarsa, F. J. Gálvez, and E. Buendía, J. Chem. Phys. 109, A. Sarsa, F. J. Gálvez, and E. Buendía, J. Chem. Phys. 109, T. Koga and H. Matsuyama, J. Chem. Phys. 107, T. Koga, J. Chem. Phys. 108, H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Twoelectron Atoms Plenum, New York, F. W. King, J. Mol. Struct.: THEOCHEM 400,

Excited states of beryllium atom from explicitly correlated wave functions.

Excited states of beryllium atom from explicitly correlated wave functions. F.J. Gálvez and E. Buendía Departamento de Física Moderna, Facultad de Ciencias, Universidad de Granada, E-87 Granada, Spain