Helium Ground State tremendously accelerated by using basis - PDF document

— — � � Atomic Wave Function Forms S. A. ALEXANDER, 1 R. L. COLDWELL 2 1 Department of Physics, University of Texas at Arlington, Arlington, Texas 76019 2 Department of Physics, University of Florida, Gainesville, Flordia 32611 Received 3 June 1996; revised 24 December 1997; accepted 2 January 1997 ABSTRACT: Using variational Monte Carlo, we compare the features of 118 trial wave function forms for selected ground and excited states of helium, lithium, and beryllium in order to determine which characteristics give the most rapid convergence toward the exact nonrelativistic energy . We find that fully antisymmetric functions are more accurate than are those which use determinants, that exponential functions are more accurate than are linear function, and that the Pade function is anomalously ´ accurate for the two-electron atom . We also find that the asymptotic and nodal behavior of the atomic wave function is best described by a minimal set of functions . � 1997 John Wiley & Sons, Inc . Int J Quant Chem 63 : 1001 � 1022, 1997 . configuration . The constant E is fixed at a value in Introduction close to the desired state in order to start the optimization in the proper region . The exact wave function is known to give both the lowest value of V puting the total energy ² : ariational Monte Carlo is a method of com- H and a zero variance . If the adjustable parame- ters in the trial wave function are optimized so as to minimize the energy, an instability often occurs . ² : Ý Ý 2 Ž . H � � H � � w � � � w 1 � This happens when a set of parameters causes i i i i i i i ² : H to be estimated a few sigma too low . Al- though such parameters will produce a large vari- Ž . and its variance i . e . , statistical error ance, they are favored by the minimization . This problem can be avoided only by using a very large 2 ½ 5 2 2 Ý Ž . 2 2 Ý 2 H � � E � � � w � � w � � number of configurations during the optimization i in i i i i i i i of the wave function so as to distinguish between Ž . 2 ² : those wave functions for which H is truly low and those which are merely estimated to be low . In using Monte Carlo integration 1 � 17 . Here, H is � � contrast, variance minimization favors those wave Ž . the Hamiltonian, � � � x is the value of the functions which have a constant local energy . Pa- i t i trial wave function at the Monte Carlo integration rameter values which do not produce this property Ž . point x , and w � w x is the relative probability will be eliminated by the optimization process . As i i i Ž of choosing this point usually referred to as a a result, only a small fixed set of configurations is needed to accurately determine the variance . Correspondence to: S . A . Alexander . � 1997 John Wiley & Sons, Inc. CCC 0020-7608 / 97 / 051001-22

ALEXANDER AND COLDWELL Previous studies have shown that the rate of convergence of a variational calculation can be Helium Ground State tremendously accelerated by using basis functions which satisfy the two-electron cusp condition and Table I presents the results of those trial wave- which have the correct asymptotic behavior function forms which consist exclusively of a prod- � 18 � 24 . Unfortunately, the integrals of such func- � uct of one-electron orbitals . These include a deter- tions can rarely be evaluated analytically . Because minant which was optimized so as to minimize the our method uses Monte Carlo integration, we can total energy � 25 , a variance-optimized determi- � easily build into the trial wave function many Ž . Ž . nant, � � � r � r , and a variance-optimized 2 1 2 features which will accelerate convergence . Al- Ž different-orbitals-for-different-spins form, � � 1 3 though, in principle, this flexibility leads to an . Ž . Ž . � P � r � r . All three forms contain enough 12 1 2 enormous number of possible forms, in practice, adjustable parameters to obtain a saturated result the ideal trial wave-function form must have a low from their respective optimization functionals . Be- variance, must add adjustable parameters in a cause � is the result of an energy minimization, it 1 straightforward manner, and must be easy to opti- is not surprising that this form has the lowest mize . energy and the largest variance . When variance In this article, we examine a variety of trial Ž . minimization, Eq . 2 , is used to optimize the ad- wave-function forms for the ground and first ex- justable parameters in � and � , the energy of 2 3 cited singlet states of helium, the triplet ground both forms increases by a significant amount while state of helium, the ground state of lithium, and their variance decreases . Even though the energy the ground state of beryllium . We use the ratio of of � is much higher than that of the Hartree � Fock 3 the variance and the number of adjustable parame- determinant, its variance is almost a factor of 2 ters to determine which forms produce the most smaller . For this reason, � will turn out to be a 3 rapid convergence . When computed at several val- better starting point for our next step which is the ues, this quantity enables us to tell whether addi- addition of correlation . tional parameters will noticeably lower the vari- In Table II, the process of including electron ance of a particular wave-function form or if this correlation begins with a study of wave-function form has saturated . All our energies and variances forms which consist of one-electron orbitals multi- are computed using a set of 4000 biased-as-ran- plied by a function of the interelectronic coordi- dom configurations which were generated specifi- Ž . nate, i . e . , g r . When the orbitals from the 12 cally for each atom 15 . In those forms which use � � Hartree � Fock determinant are used, e . g . , � , the 5 a Hylleraas or Pade-type function, we add all ´ variance drops by a factor of 2 . 9 compared to � 1 possible combinations of variables which produce and roughly 69 % of the correlation energy is ob- a given excitation level . The excitation level N tained before saturation occurs . Using a variance- denotes the sum of the exponents of the variables optimized determinant, e . g . , � , or a different- 14 in each term of the Hylleraas function, e . g . , r 2 and orbitals-for-different-spins form, e . g . , � , lowers 13 17 r r are N � 2 . Unless otherwise indicated, all the variance by an additional factor of 2 and 2 . 8, 1 23 values in this article are in atomic units . respectively . Although we find that the form of TABLE I Helium ground-state wave functions: product form. a No. ( ) Form parameters Energy au � = det 0 � 2.8655059 � 0.130 e -1 1 ( ) ( ) � = � r � r 5 � 2.7948343 � 0.111 e -1 2 1 2 ( ) ( ) ( ) � = 1 + P � r � r 10 � 2.7617876 � 0.858 e -2 3 12 1 2 [ ] Literature � 2.903724375 26 a ( ) ( ) [ ] ( ) [ Here, det = � r � r as computed by Clementi and Roetti 25 using an energy minimization. Elsewhere, � r = 1 + 1 2 1 k ] k ] 4 � � r 1 ( ) [ 4 � � r 2 Ý a r e and � r = 1 + Ý b r e . k =1 k 1 2 k =1 k 2 1002 VOL. 63, NO. 5