Probabilistic method to determine electron correlation energy

Transcription

1 Probablstc method to determne electron elaton energy T.R.S. Prasanna Department of Metallurgcal Engneerng and Materals Scence Indan Insttute of Technology, Bombay Mumba Inda A new method to determne electron elaton energy s descrbed. Ths method s based on a better representaton of the potental due to nteractng electrons that s obtaned by specfyng both the erage and standard devaton. The standard devaton s determned from a probablstc nterpretaton of the Coulomb nteracton between electrons. Ths leads to a better representaton of orbtal energes as ε ± ε, where ε s the Hartree-Fock orbtal energy and ε, the spread, s an ndcator of the magntude of elaton energy. Ths new representaton of the potental when combned wth an emprcal constant leads naturally to a new method to determne electron elaton energy. Correlaton energy s determned wthn the ndependent electron approxmaton wthout any contrbuton from hgher energy unoccuped states. A consstent physcal nterpretaton - an electron occupes a gven poston when other electrons are farther than on erage can be made. It s a general technque that can be used to determne elaton energy n any system of partcles wth nter-partcle nteracton (, r ) and can be consdered to be unversal frst step beyond mean-feld theory.

2 The problem of determnng electron elaton energy s of great mportance n quantum chemstry and sold state physcs. The orgn of ths problem can be traced to the Hartree-Fock approxmaton. It s an ndependent electron approxmaton n whch the nstantaneous electronelectron repulson, r r, s replaced by an eraged electron-electron nteracton n the Hamltonan. Consequently, the Hartree-Fock ground state has a hgher energy than the true ground state and the dfference s defned as the elaton energy. Ths problem was recognzed very early n the development of quantum mechancs as appled to molecules and solds. Subsequently, many technques he been developed to address ths problem. Confguraton Interacton, Moller-Plesset Perturbaton theory, Coupled Cluster method etc. he been developed to account for electron elaton n molecules. These methods are dscussed n detal n Refs.,. In these technques, the Hartree-Fock ground state (sngle Slater determnant) s frst determned, whch then becomes a startng pont to determne electron elaton energy. In general, t s necessary to nclude contrbutons from hgher energy unoccuped states n these methods. An mportant consequence s that the ndependent electron approxmaton s no longer vald and the physcally appealng pcture of an electron represented by ts wefuncton s lost. Addtonally, these methods are computatonally ntensve as the number of Slater determnants ncreases rapdly and are mpractcal for solds. Alternate approaches to address electron elaton wthn the sngle electron approxmaton he been made. These nclude studes based on Densty Functonal theory and on Green s functons (GW approxmaton and ts extensons). Densty Functonal Theory s an alternate method to calculate electronc structure (3-5). Wthn ths approach, the local densty approxmaton (LDA) s the standard method to ncorporate exchange effects. Many studes he been made go beyond the LDA to better characterze the exchange-elaton hole and ncorporate the effects of electron elatons (6-). The GW approxmaton and ts extensons (-5) represent another method to obtan better energes. Two broad themes can be dstngushed n these studes. The frst s to develop better functonals to descrbe electron elatons and the second s to develop schemes that lower the quas-partcle energes. In general, par-elaton functons or Green s functons play an mportant role n these studes.

3 In ths paper, a new method to determne electron elaton energy s descrbed. It s based on the fact that the potental due to nteractng electrons fluctuates at any poston. Specfyng both the erage and standard devaton - as opposed to ust the erage as n the Hartree-Fock method - better represents ths fluctuatng potental. The standard devaton can be determned from a probablstc nterpretaton of the Coulomb repulson between electrons. Startng from the Hartree-Fock approxmaton, ths method determnes the elaton energy wthn the ndependent electron approxmaton wthout any contrbuton from hgher energy unoccuped states. It results n lower orbtal energes. A consstent physcal nterpretaton - an electron occupes a gven poston when other electrons are farther than on erage can be made. It s a general technque that can be used to determne elaton energy n any system of partcles wth nter-partcle nteracton (, r ). The Hartree-Fock (HF) approxmaton s dscussed n detal n Ref.. The orbtal energy n the HF approxmaton s gven by () ε = f = h + J K () where f s the Fock operator and all symbols he ther usual meanng. The HF ground state energy s gven by E = ε ( J K ) () HF where the second term compensates for double countng and J = K. The HF ground state has a hgher energy than the true ground state partly because the Coulomb ntegral overestmates the repulson energy between two electrons. The Coulomb ntegral between two electrons s gven (n atomc unts) by ( r ) = ( r) ( r) r r = ( ) r r r r r r J d d d dr (3) Ths leads to the famlar nterpretaton that an electron n orbtal (henceforth referred to as electron ) experences an erage potental at due to electron n (electron ) gven by ( r ) = ( r ) r r dr (4) 3

4 Ths s equvalent to the expresson n classcal electrostatcs for the potental at any pont due to a contnuous charge dstrbuton, n ths case ( e) ( r ). In a classcal charge dstrbuton, the potental s constant because the charge dstrbuton s constant wth respect to tme. In the quantum mechancal case, electrons are pont partcles and only occupy varous postons wth probablty ( r ). Hence, t s readly seen that the potental at any poston s not constant but fluctuates wth ts erage gven by Eq. (4). A fluctuatng potental (or any other fluctuatng quantty) s better descrbed by specfyng ts standard devaton n addton to the erage. Ths can be acheved f a probablstc nterpretaton of Eq. (4) s made. Because ( r ) s a probablty densty functon and not a charge densty functon, the erage potental n Eq. (4) can also be nterpreted as the expectaton value of r r at. Wth ths nterpretaton, t becomes possble to determne the varance at. The varance s gven by ( r ) ( r) = dr ( ) r r Thus the varance can be determned f the expectaton value of ( ) r (5) r (frst term of Eq. (5)) can be evaluated. As the probablty densty functon s known, hgher moments can also be determned f necessary. It s now possble to better characterze the potental at electron. It can be represented as due to an ( r) = ( r) ± ( r) (6) where ( ) s the standard devaton and s gven by square root of the varance, ( ), determned from Eq. (5). In an n-electron system, the erage potental at due to n- electrons can be represented (as s well known) by the sum of erage potentals due to each electron. However, t s possble usng the method descrbed above to estmate the varance as well. The total varance can be represented by a sum of ndvdual varances and ts square root, the standard devaton, s gven by Σ' r = Σ' r = ( ) ( ) ( r ) (7) 4

5 where Σ' ndcates that the quantty has been obtaned from contrbutons of all electrons. Therefore, the potental due to the other (n-) electrons at can be represented as tot ( r) = ( r ) ± Σ'( r) Ths s a better representaton of the potental due to n- electrons at (8) than the Hartree-Fock approxmaton, whch s ust the frst term n Eq. (8). Specfyng the erage and standard devaton s the norm n descrbng any quantty that exhbts a spread n values. Eq. (8) appears to be the frst tme t has been done n the context of potental due to nteractng electrons. As electrons occupy dfferent postons, the dstance to and hence the potental at r fluctuates. r Eq. (8) accounts for the fluctuaton n the potental n a statstcal manner. Usng Eq. (8) (nstead of Eq. (4)) to calculate the potental energy of nteractng electrons leads to a better representaton of the orbtal energy as ε the HF orbtal energy obtaned from Eq. ()) and ε ±, where ε s the orbtal energy (same as ε s the spread. The method to determne ε s descrbed further below. The spread, ε, gves an estmate of the range of values about ε that orbtal energy can possess and s also an ndcator of the magntude of elaton energy. A large ε follows from a large standard devaton of the potental and mples strong fluctuatons about the erage (HF) potental. Ths suggests a sgnfcant dfference between the erage (HF) potental and the true potental, ndcatng a large value of the elaton energy. Hence, extendng the Hartree-Fock method to determne the spread of the orbtal energy, wll provde an ndcaton of the magntude of elaton energy. ε, The true value of the orbtal energy s a constant that does not exhbt any spread n the sense descrbed above. Ths s because even though the potental at (any poston) fluctuates, the true value of the potental at due to n- other electrons when electron s present s a fxed quantty that s determned by the elated moton of electrons. To determne ths true potental, t s necessary to adopt theoretcal technques startng wth the true many-body wefuncton, whch s frequently unknown. 5

6 However, usng Eq. (8) along wth an emprcal constant allows an effectve potental to be estmated that s closer to the true value of the potental than the erage (HF) potental. Ths naturally leads to a new method to determne electron elaton energy as descrbed below. Electron would prefer to occupy poston when the potental s lower than on erage as t would lower the (repulsve) energy. The effectve potental at when electron s present can be represented as eff ( r) = ( r) c ( r ) Σ'( r) The effectve potental, gven by Eq. (9), s closer to the true value of the potental at (9) when electron s present than the Hartree-Fock potental. It also mples that electron occupes poston when the other electrons are farther than on erage. The coeffcent ( r ) s a small c number multplyng the standard devaton of the total potental due to other electrons and the representaton s suffcently general. The smplest assumpton would be that of a constant value (c) for all electrons at all postons. The next assumpton would be that of a dfferent constant value for dfferent electrons (c) but ndependent of poston. Another possblty s to he one constant for electrons of same spn and another constant for electrons of opposte spn, as electrons of same spn are lkely to be farther apart due to exchange effects. The coeffcent must be chosen emprcally untl nsghts nto the nature of potental fluctuatons he been ganed. The electrostatc nteracton energy of electron due to other electrons ncludng elatons s gven by eff Σ Σ J = ( r ) ( r ) dr = J c ( r ) '( r ) ( r ) d (0) and hence, the elaton energy of electron s gven by c Σ' E = ( r ) ( r ) ( r ) d () The orbtal energy s lowered due to electron elatons and s gven by ε = ε + E () where both ε and E are negatve. The ground state energy s obtaned as cor E = ε ( J K ) E (3) gs 6

7 Hence, the total elaton energy, s gven by E = E (4) The spread of the orbtal energy,, s equal to E determned from Eq. () wth ( r ) =. ε c The hgher energy unoccuped states do not play any role n determnng electron elatons. In the HF method, the ant-symmetry of wefunctons (or Paul excluson prncple) provdes some measure of elaton among electrons of same spn resultng n an exchange hole surroundng each electron (,). In the method descrbed above, electrons of ether spn are farther than on erage (Eq. (9)) suggestng a elaton hole surroundng an electron. Ths s consstent wth the nature of Coulomb nteracton, whch s ndependent of spn. In the HF approxmaton, the electron electron nteracton term n the Hamltonan, r, s exact but the wefuncton (sngle Slater Determnant) approxmate, due to whch t becomes an eraged nteracton for the sngle electron Hamltonan. Ths shows that an exact two-partcle operator becomes an approxmate one-partcle operator when the wefuncton s approxmate. The above method can be consdered to be a ecton to ths approxmate one-partcle operator. Wthn the framework of ndependent electron approxmaton, t s equvalent to a frst-order perturbaton ecton to orbtal energes. Of all the methods to determne electron elaton energy, the Confguraton Interacton (CI) method (,) s conceptually the smplest and can be consdered to be a natural extenson of the Hartree-Fock method. Ths s because t s known that the electron-electron nteracton term r r, s exact but the wefuncton (sngle Slater Determnant) approxmate n the HF method. Therefore, the natural course of acton would be to expand the true wefuncton n a seres of determnants, n whch the HF wefuncton would be the frst term. The present method can be consdered to be another natural extenson of the Hartree-Fock method to ncorporate the effects of electron elatons. The natural course of acton after determnng the erage potental s to evaluate ts standard devaton. Combned wth an 7

8 emprcal constant, ths allows an effectve potental to be estmated that s closer to the true value than the erage (HF) potental. Therefore, ths method can also be consdered to be a natural extenson of the Hartree-Fock method. In ths method, the overestmate of the electronelectron repulson energy s ected, rather than the wefuncton as n the Confguraton Interacton method. The present method determnes electron elaton energy wthn the sngle electron approxmaton and s dfferent from the GW approxmaton (-5) and Densty Functonal approaches (6-). It does not requre any knowledge of Green s functons or par-elaton functons. It s ths conceptual smplcty that can make ths method wdely accessble and applcable. It s lmted n scope as ts obectve s to determne elaton energy rather than to provde a better theoretcal descrpton of electron elaton. Towards ths end, t requres the use of an emprcal constant. Purely theoretcal approaches to electron elatons need to od any relance on emprcal constants. To the best knowledge of the author, the method of ths paper s not to be found n exstng lterature. It s also clear that the method s general and not restrcted to electrons. In any system of partcles wth nter-partcle nteracton (,r ), the potental at any poston wll fluctuate. The frst attempt to solve the Schrodnger s equaton usually assumes that the partcle moves n an erage potental due to other partcles, whch can be called the mean-feld approxmaton. Ths paper shows that n addton to the erage, the standard devaton of the potental due to other partcles can be determned. Ths allows the spread of sngle partcle energes about ther meanfeld values to be specfed, whch provdes an ndcaton of the magntude of elaton energy. In addton, usng nformaton about the erage and standard devaton along wth an emprcal constant, an effectve potental that s closer to the true value than the erage potental can be estmated. The dfference n potental energes gves the elaton energy. Conceptually, the way beyond mean-feld theory s clear even before knowng the detals of the nteracton potental (,r ). Therefore, ths method can be consdered to be a unversal frst step beyond mean-feld theory. Emal: prasanna@tb.ac.n 8

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