学习资源量子化学课件a

1、The Hartree-Fock methodReference:A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications, Inc., New York, 1996Outline3.1 Derivation of the Hartree-Fock Equations3.2 Interpretation of Solutions to the HF Equations3.3 Restricted Cl

2、osed-Shell HF: the Roothaan Equations3.4 Model Calculations On H2 And HeH+ 3.5 Polyatomic Basis Sets 3.6 Some examples3.7 Unrestricted open-shell HF3.1 Derivation of HFWe can equate Hartree-Fock theory to a single determinant theory,and we are thus interested in finding a set of spin orbitals such t

3、hat thesingle determinant formed from these spin orbitals is the best possibledeterminantal approximation to the ground state of the N-electron system012|N According to the variational principle, the “best” spin orbitals are those which minimize the electronic energy:0001|(|)(|)2NNaabEHa h aaa bbab

4、ba 6.2 Derivation of the Hartree-Fock Equations3.1.1 Functional variation ) EHGiven any trial function, the expectation valueof the Hamiltonian operatoris a number given bySuppose we varyby an arbitrarily small amount,( HE Eis said to be a functional ofsinceits value depends on the form of a functio

5、nThe energy then becomes H HHE EEEE: the first order variation inHHE0E EIn the variation method, we are looking for thatfor whichis a minimum. ETo minimize, it is required thatBut, normally, this stationary point will be a minimum. Normally, this condition only ensures that Eany variation inis stati

6、onary with respect to.Niiic1ijjijiHccHE1An example:For a given trial wave function,We want to minimize the energysubject to the constraint thatSupplementary Material1EHLijjijijijiccEHcc1Using Lagranges method of undetermined multipliersijjiciijjijcijjiciijjijcLjijicEcEHcHcijijciijijjcijijciijijjcScE

7、ScEccjiji*ijiijijicijjijjijccESccEScjiijjjijjicijjijjijccESccESciiEScHcCSECHjjijjjij *iC is arbitrary (*iCandiCthe quantity in square brackets must be zero, i.e.,Sinceare both independent variables),3.1.2 Minimization of the energy of a single determinant 000011()()2NNNaaabEHa h aaa bbab baE011|NNaa

8、baababLEa b N210Given the single determinant,aE0 111|ababdxa baLTo minimizewith respect to the spin orbitals, subject to theconstraint that. First we construct thefunctionalof the spin orbitals,baconstitute a set of Lagrange undetermined multipliers. HereLLL 011NNabaababLEa b 011|NNaabababEb a 011|N

9、NaabababEa b . Sinceis real,abbaThus we haveaaa 111abababa bdx Let, then0111111()()()()21()()()()2NaaaaaNNaabbaabbaabbaabbabNNabbaabbaabbaabbaabEhh 11111111()()221 ()2NNNNaabbbbaaababNNaabbab ()()aabbaabb 11*()() ()NNabbabaababababbaab Using the relationshipsWe get011111()()(). .NNNNNaaaabbabbaaabab

10、EhC C babaabbaababababa b baababbaabab baabbaababab . .baababCC Also we haveAltogether, we obtain 011NNLEbaaba b 11111()(). .0NNNaaaabbabbaaabNNbaababhC C 1122221rxdJbb 122111122abbabrxdJ 1bJBy introducing the Coulomb operator, 1bKand the exchange operator 0.11111111111CCKJhxdNaNbbbaNbabbaaLLwe can

11、writeas 121211221bababKdxr 122121221bbaPdxr NbbbaaNbbbKJh1111111Na, 2, 1 1aSinceis arbitrary, it must be that f 1111bbbfhJKBy defining a Fock operatorthe above equation can be written as Nbbbaaf11111Nababbfor 122111122abbabrxdJ 121211221bababKdxrFrom the definition we see that 1111aaaaJK 1111bb abfh

12、JKThe Coulomb term: In an exact theory, the Coulomb interaction is represented by the two-electron operator rij-1 (the Coulombs law). In the Hartree or Hartree-Fock approximation, electron-1 in experiences a one-electron, local, Coulomb potentialabaNamely, the two-electron potential rij-1 felt by el

13、ectron-1 and associated with the instantaneous position of electron-2 is replaced by a one-electron potential, obtained by averaging the interaction rij-1 of electron-1 and electron-2, over all space and spin coordinates x2 of electron-2, weighted by the probability that electron-2occupies the volum

14、e element dx2 at x2. By summing over all , one obtains the total averaged potential acting on the electron in , arising from the N-1 electrons in the other spin orbitals. 22|(2)|bdxa21212212(2)(1)(1)|(2)|NNCoulabbb ab aJdxrdrrThe exchange term arises from the antisymmetry principle and is non-classi

15、cal. The exchange operator is non-local and must berepresented as an integral. Operating with oninvolves an “exchange” of electron 1 and electron 2 on the right-handside. Namely, the result of the operation depends on the value ofthroughout the whole space, not just at x1, as the Coulomb operator. (

16、1)bK(1)aaThe expectation values of the Coulomb and exchange operator are just the Coulomb and exchange integrals: (1)|(1)|(1)(|)abaJaa bb(1)|(1)|(1)(|)abaKab ba*1212(1)(1)(2)(2)(1)ababbKdxr*121212(2)2)(1)(bbaPdxr|Ncabbabf|babaf|0,( ,)raarffr sN|rsrsfThe structure of the Fock matrixwhich determines o

17、nly the N occupied spin orbitals00OOVV0OVVOFF(stationarity condition)3.1.3 The canonical HF equations 00(1) The unitarily transformed single determinantcan at most differfrom the original determinantby a global phase factor.aaof the spin orbitalsConsider a new set of spin orbitals, which is a unitar

18、y transformationbbabaUThus ddcdbbabcaUUbddbdcbabdUU acacbbcabbbcbaUUUUUUNbbbaaf1Unitary transformation of ALet us define a square matrix : NNNNANaNaNa21212122221111 ANdet!210Then, NNNNNNNNNNUUUUUUUUUNNNNNNA21222211121121212121111111AU 1detdetdetdetdetdet2UUUUUUUTherefore, ieU det00ie UAAUdetdetdet A

19、UAdetdetdet 0210detdet!UANSinceorIUUFrom, we have bbbKJhf1111f(2) The Fock operatoris invariant to a unitary transformation of thespin orbitals.bbabaUWith the transformed spin orbitals, aaaaarxdJ2211122 accacbbabUrUxd221122 bccbacabarxdUU221122 bbbbbJrxd1221122cbcbaabcaacaabacabaUUUUUUUU bbaaKK11 11

20、ffHere the following relationships are usedThus the sum of Coulomb operators is invariant to a unitary transformationof the spin orbitals.In an identical manner, we can show thatTherefore,baNbbbaaf1caNbbcbaacbcf1baabf(3) The effect of the unitary transformation oncMultiplying both sides of the above

21、 equation by, one hasorddbdbccacaUUthusLet abcddbcdaccddbcdcadccddbcabaabUUUUUUfxdUUfxd11111111UUor in matrix formaaaf*abbaUaSinceis a Hermitian matrix, it is always possible tofind a unitary matrixto diagonalize. Assume that the set of spin orbitalsdiagonalize the matrix of Lagrange multipliers, i.

22、e. ,a is called the set of canonical spin orbitals. This unique set of spin orbitalsAn infinite number of equivalent sets of orbitals can be obtained from the canonicalorbitals by unitary transformations. In particular, localized orbitals more in line withChemical intuitions can be obtained.aaaf 111

23、11 11NHFbbbfhJKhv 1111 NHFbbbvJK(4) The canonical HF equationThe canonical HF equation is 1HFv: an effective one-electron potential operator called the HF potential. 1h: a core-Hamiltonian operator.The canonical spin orbitals will generally be delocalized and form abasis for an irreducible represent

24、ation of the point group of the molecule.Namely, they have certain symmetry properties characteristic of the molecule.Nbbbaaf1Unitary transformation of 6.3 Interpretation of Solutions to the HF Equations 3.2.1 Orbital energies and Koopmans theorem |jjjf, 2, 1j 11111NbbbfhJK ijjijjijf 1Nijibjibjbfi h

25、 jJK 1()()Nbi h jij bbib bj 1|Nbi h jibjb fis a well-defined Hermitian operator.i.i: the spin orbital energy corresponding to the spin orbital occbiiiibibihif3.2 Interpretation of the HF solutions,ba,sroccupied spin orbitals:virtual spin orbitals: a: the kinetic energy and attraction of the nuclei p

26、lus a Coulomb and exchange interaction with each of the remaining (N-1) electrons in the (N-1) spin orbitals.rNbb, 2, 1exchange interaction with all N electrons (of the HF ground state) in the: the kinetic energy and attraction of the nuclei plus a Coulomb and spin orbitalsNbrNbarbrbrhrababaha110aaa

27、a1b abNaNra h aab abr h rrb rbThe virtual orbital does not resemble an excited state, but an (N+1)-electron state.NaNaNbNaaababaha)(210NaaNaNbNaababahaE0The total energy of the stateis not just the sum of the orbital energies.This is because the sum of orbital energies counts the electron-electronin

28、teractions twice.Relation between the total energy and the orbitals energies) 1(NNcccN11211(1) An approximate method for calculating ionization potentialsThe ionization process:cRemove an electron from the spin orbitalto produce the-electron state determinant state01NcNIP (ionization potential) =0Nc

29、N1Within the single determinantal approximation, and assuming that theoptimum spin orbitals inare identical with those ofabNaNNNababahaHE21000 cacbcacNcNcNababahaHE21111NcNNcccN21011211 bcacbcbcacacbcbcacacbcbcacbacbabcbcbababcbcbcacaababcbcbacacababcccccbcbacacababacacabababab2121212121212121212121

30、cbNcNcbcbchc IP01c(: the occupied spin orbital energy) The ionization potential for removing an electron from cnegative of the orbital energy c. Orbital energies aand thus ionization potentials are positive. For example,is just the are generally negative311233212|113|1323|232hhEh 2112212|12Ehh 32E12

31、3123“dressed e3”: 3“bare e3”: h33The energy required to remove electrons 1 and 2 is222112|12()E (both electrons 1 & 2 are dressed)33(1,2)ehr (2) An approximate method for calculating electron affinities) 1(NThe process of adding an electron to one of the virtual spin orbitalsto produce the-elect

32、ron stateWithin the single determinantal approximation,NNNrrN2102110NrbrNNrbrbrhr EA10The electron affinity (EA) of is The electron affinity for adding an electron to the virtual spin orbital ris just the negative of the orbital energyrrNrNEE01If r is negative, the electron affinity is positive, thu

33、s rN1 is more stable than 0N(). .In HF calculations on neutral molecules, ris almost always positive.(3) Koopmans Theorem (1933) 0NGiven an N-electron HF single determinant virtual spin orbital energies a and r, then the ionization potential to) 1(N-electron single determinant aN1with occupied and p

34、roduce anwith identical spin orbitals, obtained by removing a electron from spina, and the electron affinity to produce an ) 1(Ndeterminant rN1electron to spin orbital r, are just aandr, respectively. orbital-electron singlewith identical spin orbitals, obtained by adding an cNcNEE01IPrrNNEE10EApote

35、ntials and electron affinities. Koopmans theorem gives us a way of calculating approximate ionization) 1(NApproximations underlying the Koopmans theorem:(1) “Frozen orbital” approximation is adopted, i.e., relaxation of the spinorbitals in the-electron states is neglected here.The neglect of relaxat

36、ion tends to produce too positive an ionization potential and too negative an electron affinity. 10Koopmans101111IP IP (rela xation inc luded( 0)NNcNNccNcNNcNcEEEEEEEE Koopmans10111101EA EA (relaxatio ( 0)n included)NNNrNrNrNNrNrrEEEEEEEErelaxed(2) Correlation effects tend to cancel the relaxation e

37、rror for ionizationpotentials, but add to the relaxation error for electron affinities. This isbecause correlation energies are largest for the system with the highestnumber of electrons.In summary, Koopmans ionization potentials are reasonable firstapproximations to experimental IPs. But Koopmans E

38、As are often bad.3.2.2 Brilloins Theorem0According to the configuration interaction method, the exact ground statecan be expressed asraarraHFcc,)(000ra0arobtained from by a singlereplacement of with .: a singly excited determinantracra,0Assume that only singly excited determinants are included in th

39、e aboveexpansion, then the coefficientsare determined from the linear variationalprinciple by diagonalizing the Hamiltonian matrix in the basis of the states. ECC sbsbrasbsbsbrarararasbraHHHHHHHHH000000sbraCCCC0Using the Slater-Condon rules, )(00rafrbabrhaHarrrabraTherefore, sbsbrasbsbrararaHFHHHH00

40、00Thus, one obtains 0000 EE) (,000srbarsabrsabarraracccBrillouins Theorem: ra000raH .Singly excited determinants will not interact directly with a reference HF determinant, i.e., can mix indirectly with 0rsab by way of the matrix elements rsabH0and rsabraHthrough the doubly excited determinants In o

41、ther words, the HF ground state cannot be improved by mixing it withsingly excited determinants. However, singly excited determinantsraThe singly excited configurations are, however, important for properties such as dipole moment.6.4 Restricted Closed-Shell Hartree-Fock: the Roothaan Equations3.3.1

42、Closed-shell HF: Restricted spin orbitals rrxjjiRestricted spin orbitals:2211210NNaaNThe closed-shell restricted ground state:The spin orbital HF equation: 111xxxfiiiiijj 11111rrxfjjj)(1* 111111rrxfdjjj 11111xfdrf 111rrrfjjjMultiply both sides byand integrate, one obtains NcccxPrxxdrhxf121211222111

43、1xfBy definition, the spin orbital Fock operatoris of the form(1) The spatial Fock operator 1rf 112121122211111NcccxPrxxdrhdrfThe final form ofcan be derived as follows:spatial orbital HF equation 21211222122211112211222122211111rPrrrdddrrrrdddrhccNccc 2122112212rPrPcc(Since ) 2121122202220111221122

44、212221111rPrrrdddrrrrdddccNccc 2122112212rPrPcc(Since) 212212112222211111NcccrPrrdrddrhrf 2122121122222111NcccrPrrdrdd 22121122221122212NcccccrPrrrdrrrrdrhHence, the closed-shell spatial orbital Fock operator has the form 211112NcccKJrhrfIn contrast, the closed-shell spin orbital Fock operator has t

45、he form NcccKJrhxf1111The closed-shell HF total energy: (2) Orbital energies in the spatial orbital form: NbbibiiiihThe spin orbital expression of the orbital energy is given by /2/2/200022NNNaabEHa h aaa bbab ba22aaababaabhJK, Niiiibibbh 22NbibibiiKJh 22NNiiiibbibbiiibbbbh 20 ()()Niibbibbib 2()()Ni

46、iiibbibbibh ()()Niiiibbibbibh rxiiiiii riasi riasi, thus Letandand denote For example, in the minimal basis H2 model, one has1122 121222212122211111111111222KJhKJhJhKJh111102JhEabababaaaKJhE220The total electronic energy|0rHFaHFarHaf(Brillouin condition)Derivation with second quantization1,2tqptpqrs

47、tupqurspquH ahaagaa()()ptpqtstptspqupuspqpupqusa haga hag*uttu,tptpqtsuupuspqutH a aa hag(Koopmans operator)2bbaa42bdbddbacacac qppqpn()p qpqpqr srssrrsn n |,| ,0trtrtHFuHFururutH afff/ 212(|)(|)Npqpqafhaapqpa aq0arraff , 0orfSummary1(1)(1)(1)(1)NbbbfhJK|aaaf/21(1)(1)2(1)(1)NbbbfhJKKoopmans theorem|

48、0rHFaarHfBrilloins theorem for canonical orbitals3.3.2 Introduction of a basis: the Roothaan equations KuuuiiC1Ki, 2, 1The integro-differential HF equation can be solved with grids for atoms orlinear molecules (numerical HF), but for polyatomic molecules we usually expand the orbitals with pre-speci

49、fied basis functions (LCAO) which are expected to describe the states of interest sufficiently well. This amounts to projecting the operators to a local space spanned by the basis functions. Kuru, 2, 1 vvviivvvicCf111iSubstitutinginto the HF equation leads to vvuviivvuvirdCfrdC1111111 1uBy multiplyi

50、ng byon the left and integrating, we obtain a matrixequationDefine the following matrices:S1. The overlap matrix , 111vuuvrdS vuuvvuuvSrdrdS111111SKK 10uvSSinceis aHermitian matrix (it can be proven that) 1111vuuvfrdFF2. The Fock matrixis also aFKK Hermitian matrix.These are the Roothaan equations,

51、which can be rewritten as the singlematrix equationSCFC KKKKKKKKCCCCCCCCCCCCC2121332312222111211kOO21viviuvvjijivjuvvjjivjuvuiCSCSCSSC sinceNow the matrix equation can be written asvviuvivviuvCSCFKi, 2, 1uiuiSCFCor3.3.3 The charge Density the total charge density is just For a closed-shell molecule

52、described by a single determinant wave functionK210with each occupied MOacontaining two electrons, rardr rdra2described by the spatial wave functioninat a pointThe probability of finding an electron 2racharge density 222Naarr 22Naaarrror *( )( )( )|iijjijrrra 222 2 NvavuauavuNuavauvuvauvuvuvrCCC Crr

53、Prr KaCKuuuaa, 2, 1 1Substitutingleads tointo the above equationHere we define the density matrix (charge-density bond-order matrix) as 22()NuvuavaaPC CCnC3.3.4 Expression for the Fock matrix u In the AO basis orbital)molecular theis (Here 2122222111121111111211111a2212112211221211211NacoreuvNavaaaa

54、ucoreuvNavaauvuvNaaauvuuvavuaaauvPrrdrrdrdKJrdhrdKJhrdfrdFThe core-Hamiltonian matrix nucluvuvVvAAAuTvuvAAAuvucoreuvVTRrZrdrdRrZrdhrdnucluvuv11121112111111121112111aaCuvFInsertinginto the two-electron terms of the Fock matrixelement, one gets uvcoreuvcoreuvNaaacoreuvNaaacoreuvuvGvuuvPvuuvCCvuuvCCF 2

55、12222uvGP: the two-electron part of the Fock matrix, which depends on thedensity matrixand a set of two-electron integrals. 221111221rrdrduvvu SCCCFThe major difficulty in a Hartree-Fock calculation is the evaluation andmanipulation of these two-electron integrals. CFPFFSince, the Roothaan equations

56、 are nonlinear, i.e.,They will need to be solved in an iterative fashion3.3.5 Orthogonalization of the basis u uvvuSrrrdThe basis functions used in molecular calculations are normalized, but theyare not orthogonal to each other. For a set of basis functionsthat arenot orthogonal,vvvuuXKu, 2, 1uit is always possible to find a transformed set of functionsgiven bywhich form an orthonormal set, i.e., uvvurrrd uvvuvuvuvuvuSXXXSXXSXXrrrdXrXrXrdrrrdThus, to make the newly transformed basis functions orthonormal, it