Quantum Chemistry Notes:

Transcription

1 Quantum Chemisty otes: Hatee-Fock equations The Hatee-Fock method is the undelying appoximation to nealy all methods of computational chemisty, both ab initio and semi-empiical. Theefoe, a clea undestanding of whee it comes fom and what it means is essential fo quantum chemists. These notes outline the Hatee-Fock method and deive seveal of the most useful HF equations. These notes ae oganized as follows: I) Geneal pinciples and definitions. II) Enegy expession fo a Slate deteminant wave function III) The geneal Hatee-Fock equations. IV) The Roothan Equations fo closed shell molecules I. Geneal pinciples and definitions A) Wavefunctions and Slate deteminants: In quantum chemisty, one is usually inteested in solving the molecula Schödinge equation: HΨ=EΨ (Eq. ) whee H is the non-elativistic, Bon-Oppenheime, molecula Hamiltonian (in atomic units): H = 2 i 2 M Z A A= ia + j>i ij M M Z A Z B (Eq. 2) B>A AB B= whee i and j ae electon indices, A and B ae a nuclei indices, Z is the nuclea chage, is the numbe of electons, and M is the numbe of nuclei. The final tem in (Eq. 2) is the nuclea epulsion enegy which is a function only of the nuclea coodinates and will be ignoed in the subsequent discussion. Fo convenience, H is usually divided into a one-electon tem (h(i)) and a two-electon tem (g(i,j)) defined as follows: M hi = 2 2 Z i A gi,j = ij so that A= ia H = hi + gi,j (Eq. 3) j>i

2 In the Hatee-Fock method, the electonic wavefunction is usually appoximated as a single Slate deteminant (vide infa): Ψ HF ( x,,..., )= x ψ 2 ( x ) L ψ ( x ) ψ 2 L ψ ψ x ψ M M O M ψ x ψ 2 L ψ x x (Eq. 4) which is abbeviated as: Ψ HF ( x,,..., x )=! det ψ ( x )ψ 2 ( )...ψ x whee each is a othonomal, single electon spin obital which actually is a poduct of a spatial obital and a spin function (s)=(s) o (s). These functions have the following popeties: ψ i ψ j =δ ij αα s s = ββ s = s αβ s = s βα s = s 0 (Eq. 5) The mathematical popeties of the Slate deteminant guaantee two closely-elated popeties that ae necessay fo any multielecton wavefunction: antisymmety and the Pauli pinciple. The fist states that swapping the positions and spins of any two electon simply changes the sign of the wavefunction. This coesponds to swapping two ows of the deteminant which changes its sign. The Pauli pinciple states that two electons of identical spin cannot be at the exact same point in space. This would coespond to two ows of the deteminant being equal, which would make the oveall deteminant equal zeo, (hence meaning that the pobability of this happening is zeo.) It is impotant to emembe that the Slate deteminant given in (eq. 4) is not a matix, but a deteminant that can be expanded to yield a sum of poducts of spin obitals. Moe pecisely, the deteminant expansion contains tems fo all possible pemutations of indices (x i ) in the spinobitals: Ψ=! detψ ( x )ψ 2 ( )...ψ ( x )=! ( ) P i P! i ψ ( x )ψ 2 ( )...ψ =! ψ ( ( x )ψ 2 ( )...ψ ( x )) ( ψ ( x )ψ 2 ( )...ψ ( x ))L ( x (Eq. 6) whee P i is the pemutation opeato that pemutes the indices in all! possible combinations. B) The Method of Lagange Undetemined Multiplies: A functional is just a function that takes an function as an agument and etuns a value. Fo example, conside the functional F and the function g:

3 If F[ g x ] dxg x and g x then F[ g x ]= 2 π 0 e x2 Functional deivatives can be defined, analogous to the deivative of egula functions, that specify how the value of a functional vaies as its agument vaies. These deivatives can be used to find the functions that minimize a paticula functional. Sometimes it is desiable to find the function that minimizes a functional, but with an additional constaint. Fo example, we might want to find the function, ƒ, that minimizes F[ƒ], but with the additional condition that G[ƒ]=n (whee G is anothe functional and n is a numbe). One method fo accomplishing this is called the method of Lagange undetemined multiplies. This involves constucting a function, L, which includes both functionals and a tem, which is the undetemined multiplie: Lf,λ = Ff λ Gf n Recall that the constaint on ƒ is that G[ƒ]=n, so the tem in paenthesis is zeo and theefoe the minimum of L occus fo exactly the same ƒ as fo F. Hence we simply need to set the fist vaiation of f to 0: = 0 δl( f,λ)= δf[ f] λ δf[ f] then solving fo ƒ in tems of and then solving fo by imposing the oiginal constaint, G[ƒ]=n. II. Enegy expession fo Slate Deteminant: The quantum mechanical expectation value fo the total molecula enegy is given by: E = Ψ H Ψ ΨΨ (Eq. 7) whee H is the Hamiltonian given in Eq. 2. To calculate the enegy of a single Slate deteminant wavefunction, we simply substitute Eq. 3 into this expession (note that the denominato of Eq. 7 equals since the Slate deteminant is nomalized): E =! det ψ ( x )ψ 2 ( )...ψ ( x )hi! det ψ ( x )ψ 2 ( )...ψ x (Eq. 8) To evaluate Eq. 8, we will sepaately teat fist the one-electon and then the two-electon tems in the Hamiltonian. a) One-electon tems:

4 E = + j>i! det ψ ( x )ψ 2 ( )...ψ ( x )hi! det ψ ( x )ψ 2 ( )...ψ ( x )gi,j! detψ ( x )ψ 2 ( )...ψ! det ψ x ( x )ψ 2 ( )...ψ x Eq. 9 If we simply expand each Slate deteminant, each integal will be a sum of! x! tems. The key to educing this expession is to use the nomalization conditions given in Eq. 5 and the fact that h(i) only opeates on the i th electon coodinate, so that all othe spin obitals can be factoed out into sepaate integations. Using the Geek subscipts to indicate an abitay pemutation of the spin obitals, the value of any single tem in the! x! expansion has the following value (see next): ψ α ( x )ψ β ( )...ψ γ ( x )hi ψ χ ( x )ψ δ ( )...ψ ε ( x ) = ψ µ ( x i )hi ψ ν ( x i ) ψ α ( x )ψ β ( )...ψ γ ( x )ψ χ ( x )ψ δ ( )...ψ ε whee ψ µ x i and ψ ν do not appea in the second tem above. = ψ µ ( x i )hi ψ ν ( x i ) ψ α ( x )ψ χ = ψ µ ( x i )hi ψ ν ( x i )δ αχ δ βδ L δ γε x i x x ψ β ψ δ L ψ γ ψ ε This means that the expession will be non - zeo only when and have the same electon coodinate, etc. fo all obital ovelaps. Hence, out of the! x! tems in an h matix element, only ( - )! will be non-zeo, since that's how many ways - coodinates can be distibuted into - ovelap integals. Thus, the h matix element is: ψ α ( x )ψ β ( )...ψ γ ( x )hi ψ χ ( x )ψ δ ( )...ψ ε = ψ! x ( i )hi ψ x i + ψ 2 x ( i )hi ψ 2 x i ( x ) + L + ψ x ( ( i )hi ψ ( x )) i This esult is the same fo each of the h(i) matix elements in the expession fo E, so that the one-electon enegy fo a Slate deteminant wavefunction is: E = x x ψ i h ψ i (Eq. 0) x (ote that the sum is now ove spin obitals athe than electon indices.) x b) Two-electon tems: E 2 =! det ψ ( x )ψ 2 ( )...ψ ( x )gi,j! det ψ ( x )ψ 2 ( )...ψ ( x ) (Eq. ) j>i

5 ow, following a vey simila appoach to that we took fo the one-electon tems, we conside an abitay tem fom the! x! tems in the g(i,j) matix element: ψ α ( x )ψ β ( )...ψ γ ( x )gi,j ψ χ ( x )ψ δ ( )...ψ ε ( x ) = ψ µ ( x i )ψ λ gi,j x j ψ ν ( x i )ψ σ x j ( x )ψ β ( )...ψ γ ( x )ψ χ ( x )ψ δ ( )...ψ ε = ψ µ ( x i )ψ λ gi,j x j ψ ν ψ σ x i ψ α x j ψ α ψ χ x ψ β x ψ δ = ψ µ ( x i )ψ λ gi,j x j ψ ν ( x i )ψ σ δ x j αχ δ βδ L δ γε the poduct of delta functions will be non - zeo in only two situations: (i) = and =, o (ii) = and =, so: = ψ µ ( x i )ψ ν gi,j x j ψ µ ( x i )ψ ν o = ψ µ ( x i )ψ ν gi,j x j ψ ν ( x i )ψ µ δ x j αχ δ βδ L δ γε δ x j αχ δ βδ L δ γε L ψ γ x x ψ ε In the! x! expansion fo each g(i,j) above, thee will be (-)! tems of type (i) and (-)! tems of type (ii). In ode to detemine the sign of the abitay matix elements deived above, we need to look caefully at the signs of the individual tems in the Slate deteminant expansion. The final esult is that the fist tem above has a positive sign and the second a negative sign. Hence, the esulting tems fo each g(i,j) ae:! detψ ( x )ψ 2 ( )...ψ ( x )gi,j! det ψ ( x )ψ 2 ( )...ψ ( x ) ψ ( x i )ψ 2 gi,j x j ψ ( x i )ψ 2 x j ψ ( x i )ψ 2 gi,j x j ψ 2 ( x i )ψ x j + = ψ! ( x i )ψ 3 gi,j x j ψ ( x i )ψ 3 x j ψ ( x i )ψ 3 gi,j x j ψ 3 ( x i )ψ x j + L ψ ( x i )ψ gi,j x j ψ ( x i )ψ x j ψ ( x i )ψ gi,j x j ψ ( x i )ψ x j Thus, we get the following expession fo the two-electon enegy in tems of spin obitals: E 2 = d x d * ψ i ( x )ψ i ( x ) * ψ j ( )ψ j ( ) 2 2 dx dx * 2 ψ i ( x )ψ j ( x ) * ψ (Eq. 2) j ( )ψ i ( ) 2 ote that the tem in backets is simply the diffeence between the two tems, not a vecto. Combining Eq. 0 and Eq. 2, we get ou completed enegy expession fo a Slate deteminant wave function: x

6 E[ { ψ i }]= dx * ψ i ( x )h( x )ψ j ( x ) + dx d * ψ i ( x )ψ i x 2 d x d * ψ i = i ( x )ψ j defining the following notation fo integations ove spin obitals: [ ihj] dx * ψ i ( x )h ( x )ψ j ( x ) [ ijkl] dx dx * 2 ψ i ( x )ψ i ( x ) * ψ k ( )ψ l 2 x * ψ j ( )ψ j 2 * ψ j ( )ψ i 2 we get the following final expession fo the enegy of a Slate deteminant of spin obitals: E[ { ψ i }]= [ ihi] + [ ii jj] [ ij ji] (Eq. 3) 2 c) Slate's ules: This can be genealized fo the calculation of matix elements of one- and two-electon opeatos between deteminantal wave functions. (i) One-electon opeatos Conside the matix elements A FB of a one-electon opeato F = f(i) between the two Slate deteminants: A = a a 2...a i...a * and B = b b 2...b i...b. We get, thus, the following ule: (ii) 0 if A B by moe than one spinobital A FB = ± a k fb l if A = a...a k a k a k +...a and B = a...a k b l a k +...a a i fb i if A = B Two-electons opeatos Conside the matix elements A GB of a two-electons opeato G = g(i, j) between the * two Slate deteminants: A = a a 2...a i...a following ule: j>i and B = b b 2...b i...b. We get, thus, the

7 0 if A B by moe than two spinobitals ± [ a k b m ga l b n a k a l gb m b n ] if A B by exactly 2 spinobitals{ a k a l }and b m b n A GB = ± [ a k b t ga t a t a k a t ga t b l ] if A B by exactly spinobital a k and b l t [ a k a k ga t a t a k a t ga k a t ] if A = B k>t { } III. Hatee-Fock Equations: Given the final enegy expession deived in section II (Eq. 3), we need to deive the equations necessay to find the optimal spin-obitals to vaiationally minimize the enegy. Stating with the final enegy expession given above (using the [ij kl] integal nomenclatue defined in section II), we want to minimize E[{ i }], subject to the constaint that the spin-obitals ae othonomal: E[ { ψ i }]= [ ihi] + 2 [ ii jj] ij ji with the constaint: [i j] = ij Using the method of Lagange undetemined multiplies fo optimizing a functional with constaints (see section I), we define: L[ { ψ i }]= E[ { ψ i }] ε ij [] δ ij ij whee ij ae the 2 undetemined multiplies. We want to find the { i } that minimize E, so we want to find wee L = 0: =δe[{ ψ i }] ε ij δ[] ij =0 (Eq. 4) δl { ψ i } Expanding the vaiations in the two tems on the ight hand side of equation 4:

8 =δ [ ihi] + δ [ ii jj] [ ij ji] i = 2 = [ δψ i h ψ i ]+[ ψ i h δψ i ] δe { ψ i } [ δψ i ψ i ψ j ψ j ] + ψ i δψ i ψ j ψ j i = [ δψ i ψ j ψ j ψ i ] + ψ i δψ j ψ j ψ i +ψ [ i ψ i δψ j ψ j] + [ ψ i ψ i ψ j δψ j] +ψ [ i ψ j δψ j ψ i] + [ ψ i ψ j ψ j δψ i] Fom the definition of the integal nomenclatue, we have the following equivalencies: [i j] = [j i]* [i h j] = [j h i]* [ij kl] = [kl ij] [ij kl] = [ji lk]* Using these elations, the expession fo E educes to: = [ δψ i h ψ i ] complex conjugate δe { ψ i } Similaly, fo the second tem in Eq. 4: [] [ δψ i ψ i ψ j ψ j ] +δψ i ψ j ψ j ψ i ε ij δ ij = ε ij δij + ε ij i δj i = = ε ij δij + ε * ij δij * = ε ij δij + complex conjugate Befoe combining the diffeent tems in the vaiation, let s define two opeatos fo convenience, note that they ae defined in tems of thei esult when opeating on a spin obital. J i ψ j = d K i ψ j = d ψ * i 2 ψ i 2 2 ψ j ψ * i 2 ψ j 2 2 ψ i

9 J() is known as the Coulomb opeato and K() is known as the Exchange opeato. ote that K() is a non-local opeato since the esult of its opeation on an obital depends on the value of that obital thoughout all of space, and not just at point x. Substituting these definitions and accumulating both tems in the expession fo the vaiation, we get: δe[ { ψ i }]= d x δψ * i h + complex conjugate = 0 ψ i ψ i + J j K j ε ij ψ j The vaiation in the obital,, is completely abitay, so that in ode fo the above expession to always be equal to zeo, the tem in the squae backets must be zeo fo all values of i. Slightly eaanging the tems in the backets we get: h + J j K j ψ i= ε ij ψ j fo, 2, 3, The tem in squae backets in this expession is known as the Fock opeato, ƒ(). Substituting this we get a seies of integal equations known as the Hatee-Fock equations: f ψ i = ε ij ψ j o in matix fom: f ψ fo, 2, 3, (Eq. 5) = ψ ε (Eq. 6) whee ψ is a vecto of spin obitals and is the matix of Lagange multiplies. ote that we etain consideable flexibility in the choice of the Lagange undetemined multiplies { ij } intoduced to insue that the spin obitals ae othonomal. To see this, conside a new vecto of spin obitals elated to by a unitay tansfomation T: θ T = ψ. Substituting into the Hatee-Fock equations: = εψ θ T = θ f ψ f Tε multiplying both sides of the equation by T : f θ TT = θ ( ) TεT f θ = θ TεT sin ce TT = We have complete feedom to choose T, so that we can choose T to be the unitay matix that diagonalizes. Using this choice of T, leads us to one paticula case of the Hatee-Fock equations, known as the Canonical Hatee-Fock Equations:

10 f ψ i = ε ij ψ j fo, 2, 3, but, since we' e using a diagonalized : (Eq. 7) f ψ i =ε ii ψ i ( ) fo, 2, 3, It is impotant to emembe that the set of spin obitals solved fo in the above equation ae only one set of an infinite numbe of valid spin obitals that would aise fom diffeent choices of the tansfomation matix T, all yielding the same enegy. Anothe point to emembe, that will be clea when we deive the Roothan equations, is that the Fock opeato, ƒ(), is itself a function of all spin obitals. This means that the Hatee-Fock equations must be solved iteatively. IV. Roothan Equations fo closed shell molecules: The Hatee-Fock equations deived above make no assumption about the functional fom of the individual spin obitals. In fact, fo simple systems, these equations can be solved by standad methods fo numeically solving integal equations. Howeve, a geneally moe useful appoach is to ecast these integal equations as matix equations. The fist step is to convet the canonical Hatee-Fock equations into a spin-independent fom (we ll only do this fo closed-shell molecules). Afte that, we ll substitute as basis set expansion fo the spatial pat of the obitals to geneate the Roothan matix equations which ae at the heat of vitually all Hatee-Fock pogams. A) Closed shell HF equations: In closed shell molecules the electons ae all paied into /2 obitals. The spatial potion of a given pai of such obitals is identical: whee φ i Using the nomenclatue: = φ iα( ω) φ i βω ψ i x ae the spatial obitals, φ i =φ i αω φ i =φ i βω The closed shell gound state can be witten: is the 3D spatial coodinate, and is the spin coodinate. Ψ 0 = ψ ψ 2...ψ ψ =φ φ...φ 2 φ 2 We want the convet the geneal HF equation to /2 equations fo the spatial obitals {}. To do this we must integate the equations ove the spin coodinate. Conside a single one of the geneal HF equations:

11 f ( x )ψ i ( x )= ε i ψ i x assuming i is an spin obital: f ( x )φ i αω ( )=ε i φ i ( )α ω (Eq. 8) multiplying on the left by a α * ( ω ) and integating ove : dω α * ( ω )f x [ ( )αω ( )]φ i =ε i φ i Evaluating the closed-shell spin-independent Fock opeato ƒ( ): [ dω α * ( ω )f ( x )αω ( )] φ i dω α * ( ω ) h [ dω α * ( ω )h αω ] φ i = ( ) + J j K j + dω αω ( ) φ i = α * ( ω ) ( J j K j ) αω ( ) φ i We go ahead and integate the one - electon tem and sepaate the sums in the two electon tem into sums ove /2 and electons: φ i = hφ i f 2 + dω dω 2 d 2 α * * ( ω )φ j ( 2 )α * ω dω dω 2 d 2 α * * ( ω )φ j ( 2 )β * ω dω dω 2 d 2 α * ω dω dω 2 d 2 α * ω * φ j ( 2 )α * ω 2 * φ j ( 2 )β * ω 2 φ j ( 2 )αω ( 2 )αω ( )φ i 2 φ j βω 2 ( 2 )αω ( )φ i 2 φ j αω ( )αω ( 2 )φ i 2 φ j βω ( )αω ( 2 )φ i 2 ow we can go ahead and do the spin integations. ote that all of these integations equal, except fo the last tem which integates to zeo. This leaves us with the following expession fo the closed shell, spin-independent Fock matix: φ i = hφ i f 2 +2 d * 2 φ j ( 2 ) 2 φ j φ 2 i d * 2 φ j φ j 2 2 φ i 2

12 we can ewite this as: φ i = hφ i + 2J j K j f 2 ( )φ i whee J and K ae the closed shell coulomb and exchange opeatos: J j φ i = d * 2 φ j 2 φ j ( 2 ) 2 φ i and K j φ i = d * 2 φ j ( 2 ) φ i ( 2 ) 2 φ j ow, combining this esult with the oiginal ight hand side of the HF equations (in Eq. 8), we get the following closed shell HF equations: φ i f =ε i φ i fo i =, 2,3,, /2 (Eq. 9) Following the same pocedue, we can ewite the HF enegy expession fo a closed shell molecule: 2 E = 2 ( ih i) + 2iijj) ij ji) ( = 2 h ii + 2J ij K ij 2 2 i = whee (Eq. 20) ( J ij d * 2 φ i J 2 j ( 2 )φ i 2 K ij d * 2 φ i K j φ i B) Intoduction of a basis set: In ode to make the closed shell HF equations solvable using matix opeations, it s necessay to intoduce a basis set to appoximate the individual spin obitals. This basis set {} is usually a set of catesian gaussian functions (see lectue notes), but the equations below ae geneal fo any type of basis function: = C ai χ a φ i basis a= Substituting this expansion into the closed shell HF equations (see next): f f φ i =ε i φ i basis C νi χ ν ν= basis =ε i C νi χ ν ν=

13 To convet to a matix equation, multiply on the left by χ µ ( ) and integate: basis C νi ν= d χ µ fχ ν = ε i C νi d χ µ basis ν= χ ν ow, defining the Fock matix F, and the ovelap matix S: F µν = d χ µ fχ ν and S µν = d χ µ χ ν we get the Roothan matix equations fo closed shell molecules: basis F µν C νi = ε i S µν C νi ν= basis ν= o in pue matix fom: fo i =,2,..., basis FC = SC whee is a basis basis diagonal matix. The Roothan equation deived above cannot be solved as a simple matix eigenvalue equation (AX=X) because the ovelap matix S, appeas on the ight hand side. We can get aound this poblem by making a substitution fo the C matix in the Roothan matix equation: Stating with the matix Roothan equations: FC = SC make the following substitution: C' = XC to yield: C = X - C' FXC' = SXC' Multiplying on the left by X : X FXC' = X SXC' If we choose X to be a unitay matix that diagonalizes S, such as S /2, then X SX = F'C' = C' ( unit matix), so that the equations become: (defining F' = X FX) This final expession fo the closed shell Roothan matix equations ae what ae actually implemented in conventional HF pogams. This equation shows how to deive a new C matix fom the Fock matix by solving a matix eigenvecto poblem. But, as we stated ealie, the Fock matix is itself a function of the obitals with means that it is a function of C. Hence we still equie a convenient expession fo constucting the Fock matix fom C.

14 F µν = d χ µ f χ ν ( ) = d 2 χ µ + d χ µ hχ ν i = [ ]χ ν 2J i K i Substituting the basis set expansion into the definitions of J and K and enaming the one electon potion of the Fock matix: coe = H µν + coe = H µν + whee: P λσ = and: 2 2 basis basis i λ σ basis basis λ C λi C σi i ( µν λσ) d d * 2 χ µ σ Cλi C σi 2 ( µνλσ) ( µσλν) Pλσ 2 ( µνλσ) ( µσλν) * χ ν 2 2 χ λ χ σ 2 This final set of definitions, combined with the matix eigenvalue equation, F C = C, ae a complete set of expession fo witing a closed shell Hatee Fock pogam. ote that the two sets of integals defined above, H coe, and ( ) ae calculated using numeical integation schemes that depend on the exact fom of the individual basis functions {}. Specific implementations of these numeical integation methods, and many complete HF packages ae available fom the commecial vendos and fom the QCPE.