N.18 Cor­re­la­tion en­ergy

The er­ror in Hartree-Fock is due to the sin­gle-de­ter­mi­nant ap­prox­i­ma­tion only. A term like “Hartree-Fock er­ror“ or sin­gle-de­ter­mi­nan­tal er­ror is there­fore both pre­cise, and im­me­di­ately un­der­stand­able by a gen­eral au­di­ence.

How­ever, it is called cor­re­la­tion en­ergy, and to jus­tify that term, it would have to be both clearer and equally cor­rect math­e­mat­i­cally. It fails both re­quire­ments mis­er­ably. The term cor­re­la­tion en­ergy is clearly con­fus­ing and dis­tract­ing for non­spe­cial­ist. But in ad­di­tion, there does not seem to be any the­o­rem that proves that an in­de­pen­dently de­fined cor­re­la­tion en­ergy is iden­ti­cal to the Hartree-Fock sin­gle de­ter­mi­nant er­ror. That would not just make the term cor­re­la­tion en­ergy disin­gen­u­ous, it would make it wrong.

In­stead of find­ing a rig­or­ous the­o­rem, you are lucky if stan­dard text­books, e.g,. [30,34,46] and typ­i­cal web ref­er­ences, of­fer a vague qual­i­ta­tive story why Hartree-Fock un­der­es­ti­mates the re­pul­sions if a pair of elec­trons gets very close. That is a symp­tom of the dis­ease of hav­ing an in­com­plete func­tion rep­re­sen­ta­tion, it is not the dis­ease it­self. Low-pa­ra­me­ter func­tion rep­re­sen­ta­tions have gen­eral dif­fi­culty with rep­re­sent­ing lo­cal­ized ef­fects, what­ever their phys­i­cal source. If you make up a sys­tem where the Coulomb force van­ishes both at short and at long dis­tance, such cor­re­la­tions do not ex­ist, and Hartree-Fock would still have a fi­nite er­ror.

The ki­netic en­ergy is not cor­rect ei­ther; what is the cor­re­la­tion in that? Some sources, like [30] and web sources, seem to sug­gest that these are in­di­rect re­sults of hav­ing the wrong cor­re­la­tion en­ergy, what­ever cor­re­la­tion en­ergy may be. The idea is ap­par­ently, if you would have the elec­tron-elec­tron re­pul­sions ex­act, you would com­pute the cor­rect ki­netic en­ergy too. That is just like say­ing, if you com­puted the cor­rect ki­netic en­ergy term, you would com­pute the cor­rect po­ten­tial too, so let’s re­name the Hartree-Fock er­ror “ki­netic en­ergy in­ter­ac­tion.”

Even if you com­puted the po­ten­tial en­ergy cor­rectly, you would still have to con­vert the wave func­tion to sin­gle-de­ter­mi­nan­tal form be­fore eval­u­at­ing the ki­netic en­ergy, oth­er­wise it is not Hartree-Fock, and that would pro­duce a fi­nite er­ror. Phrased dif­fer­ently, there is ab­solutely no way to get a gen­eral wave func­tion cor­rect with a fi­nite num­ber of sin­gle-elec­tron func­tions, what­ever cor­rec­tions you make to the po­ten­tial en­ergy.

Sz­abo and Ostlund [46, p. 51ff,61] state that it is called cor­re­la­tion en­ergy since “the mo­tion of elec­trons with op­po­site spins is not cor­re­lated within the Hartree-Fock ap­prox­i­ma­tion.” That is in­com­pre­hen­si­ble, for one thing since it seems to sug­gest that Hartree-Fock is ex­act for ex­cited states with all elec­trons in the same spin state, which would be lu­di­crous. In ad­di­tion, the elec­trons do not have mo­tion; a sta­tion­ary wave func­tion is com­puted, and they do not have spin; all elec­trons oc­cupy all the states, spin up and down. It is the or­bitals that have spin, and the spin-up and spin-down or­bitals are most def­i­nitely cor­re­lated.

How­ever, the au­thors do of­fer a clar­i­fi­ca­tion; they take a Slater de­ter­mi­nant of two op­po­site spin or­bitals, com­pute the prob­a­bil­ity of find­ing the two elec­trons at given po­si­tions and find that it is cor­re­lated (un­less the spa­tial or­bitals are equal). They then de­clare that these cor­re­lated po­si­tions mean that the mo­tion of the two elec­trons is uncor­re­lated.

The un­re­stricted Hartree-Fock so­lu­tion of the dis­so­ci­ated hy­dro­gen mol­e­cule is of this type. This so­lu­tion was dis­cussed in the in­tro­duc­tory sec­tion 9.3.1. Since if one elec­tron is around the left pro­ton, the other is around the right one, and vice versa, nor­mal peo­ple would call the po­si­tions of the elec­trons strongly cor­re­lated. And even while mo­tion is not de­fined on quan­tum scales, to any en­gi­neer it would seem lu­di­crous to claim that strongly cor­re­lated po­si­tions would pro­duce un­cor­re­lated mo­tion” if “mo­tion ex­isted.

Koch and Holthausen, [30, pp.22-23], ad­dress the same two elec­tron ex­am­ple as Sz­abo and Ostlund, but do not have the same prob­lem of find­ing the elec­tron prob­a­bil­i­ties cor­re­lated. For ex­am­ple, if the spin-in­de­pen­dent prob­a­bil­ity of find­ing the elec­trons at po­si­tions ${\skew0\vec r}_1$ and ${\skew0\vec r}_2$ in the dis­so­ci­ated hy­dro­gen mol­e­cule is

$${\textstyle\frac{1}{2}}\vert\psi_{\rm {l}}({\skew0\vec r}_1... ...0\vec r}_1)\vert^2\vert\psi_{\rm {l}}({\skew0\vec r}_2)\vert^2$$

then, Koch and Holthausen ex­plain to us, the sec­ond term must be the same as the first. Af­ter all, if the two terms were dif­fer­ent, the elec­trons would be dis­tin­guish­able: elec­tron 1 would be the one that se­lected $\psi_{\rm {l}}$ in the first term that Koch and Holthausen wrote down in their book. So, the au­thors con­clude, the sec­ond term above is the same as the first, mak­ing the prob­a­bil­ity of find­ing the elec­trons equal to twice the first term, $\vert\psi_{\rm {l}}({\skew0\vec r}_1)\vert^2\vert\psi_{rm{r}}({\skew0\vec r}_2)\vert^2$. That is an un­cor­re­lated prod­uct prob­a­bil­ity.

How­ever, the as­sump­tion that elec­trons are in­dis­tin­guish­able with re­spect to math­e­mat­i­cal for­mu­lae in books is highly con­tro­ver­sial. Many re­spected ref­er­ences, and this book too, only see an em­pir­i­cal re­quire­ment that the wave func­tion, not books, is an­ti­sym­met­ric with re­spect to ex­change of any two elec­trons. And the wave func­tion is an­ti­sym­met­ric even when the two terms above are not the same.

Wikipedia, [[21]], Hartree-Fock en­try June 2007, lists elec­tron cor­re­la­tion, (de­fined here vaguely as ef­fects aris­ing from from the mean-field ap­prox­i­ma­tion, i.e. us­ing the same $v^{\rm {HF}}$ op­er­a­tor for all elec­trons) as an ap­prox­i­ma­tion made in ad­di­tion to us­ing a sin­gle Slater de­ter­mi­nant. Sorry, but Hartree-Fock gives the best sin­gle-de­ter­mi­nan­tal ap­prox­i­ma­tion; there is no ad­di­tional ap­prox­i­ma­tion made. The mean field ap­prox­i­ma­tion is a con­se­quence of the sin­gle de­ter­mi­nant, not an ad­di­tional ap­prox­i­ma­tion. Then this ref­er­ence pro­ceeds to de­clare this cor­re­la­tion en­ergy the most im­por­tant of the set, in other words, more im­por­tant that the sin­gle-de­ter­mi­nant ap­prox­i­ma­tion! And again, even if the po­ten­tial en­ergy was com­puted ex­actly, in­stead of us­ing the $v^{\rm {HF}}$ op­er­a­tor, and only the ki­netic en­ergy was com­puted us­ing a Slater de­ter­mi­nant, there would still be a fi­nite er­ror. It would there­fore ap­pear then that the name cor­re­la­tion en­ergy is suf­fi­ciently im­pen­e­tra­ble and poorly de­fined that even the ex­perts can­not nec­es­sar­ily fig­ure it out.

Con­sider for a sec­ond the ground state of two elec­trons around a mas­sive nu­cleus. Be­cause of the strength of the nu­cleus, the Coulomb in­ter­ac­tion be­tween the two elec­trons can to first ap­prox­i­ma­tion be ig­nored. A reader of the var­i­ous vague qual­i­ta­tive sto­ries listed above may then be for­given for as­sum­ing that Hartree-Fock should not have any er­ror. But only the un­re­stricted Hartree-Fock so­lu­tion with those nasty, un­cor­re­lated (true in this case), op­po­site-spin elec­trons (or­bitals) is the one that gets the en­ergy right. A un­re­stricted so­lu­tion in terms of those per­fect, cor­re­lated, aligned-spin elec­trons gets the en­ergy all wrong, since one or­bital will have to be an ex­cited one. In short the cor­re­la­tion en­ergy (er­ror in en­ergy) that, we are told, is due to the mo­tion of elec­trons of op­po­site spins not be­ing cor­re­lated is in this case 100% due to the mo­tion of aligned-spin or­bitals be­ing cor­re­lated. Note that both so­lu­tions get the spin wrong, but we are talk­ing about en­ergy.

And what hap­pened to the word er­ror in cor­re­la­tion en­ergy er­ror? If you did a fi­nite dif­fer­ence or fi­nite el­e­ment com­pu­ta­tion of the ground state, you would not call the er­ror in en­ergy trun­ca­tion en­ergy; it would be called trun­ca­tion er­ror or “en­ergy trun­ca­tion er­ror.” Why does one sus­pect that the ap­pro­pri­ate and in­for­ma­tive word er­ror did not sound hot enough to the physi­cists in­volved?

Many sources re­fer to a ref­er­ence, (Löwdin, P.-E., 1959, Adv. Chem. Phys., 2, 207) in­stead of pro­vid­ing a solid jus­ti­fi­ca­tion of this widely-used key term them­selves. If one takes the trou­ble to look up the ref­er­ence, does one find a rig­or­ously de­fined cor­re­la­tion en­ergy and a proof it is iden­ti­cal in mag­ni­tude to the Hartree-Fock er­ror?

Not ex­actly. One finds a vague qual­i­ta­tive story about some per­ceived holes whose math­e­mat­i­cally rig­or­ous de­f­i­n­i­tion re­mains re­stricted to the cen­ter point of one of them. How­ever, the lack of a de­fined hole size is not sup­posed to de­ter the reader from agree­ing whole­heart­edly with all sorts of claims about the size of their ef­fects. Terms like main er­ror,, “small er­ror,” large cor­re­la­tion er­ror (qual­i­fied by cer­tainly”), “van­ish or be very small, (your choice), are bandied around, even though there is no small pa­ra­me­ter that would al­low any rig­or­ous math­e­mat­i­cal de­f­i­n­i­tion of small or big.

Then the au­thor, who has al­ready noted ear­lier that the ref­er­ences can­not agree on what the heck cor­re­la­tion en­ergy is sup­posed to mean in the first place, states “In or­der to get at least a for­mal de­f­i­n­i­tion of the prob­lem, ...” and pro­ceeds to re­de­fine the Hartree-Fock er­ror to be the “cor­re­la­tion en­ergy.” In other words, since cor­re­la­tion en­ergy at this time seems to be a pseudo-sci­en­tific con­cept, let’s just cross out the cor­rect name Hartree-Fock er­ror, and write in “cor­re­la­tion en­ergy!”

To this au­thor’s credit, he does keep the word er­ror in cor­re­la­tion er­ror in the wave func­tion in­stead of us­ing cor­re­la­tion wave func­tion. But some­how, that par­tic­u­lar term does not seem to be cited much in lit­er­a­ture.