The Hartree-Fock approximation

Contents The Born-Oppenheimer approximation Literature Quantum mechanics 2 - Lecture 7 November 21, 2012

Contents The Born-Oppenheimer approximation Literature 1 The Born-Oppenheimer approximation 2 3 Literature

Contents The Born-Oppenheimer approximation Literature Contents 1 The Born-Oppenheimer approximation 2 3 Literature

Contents The Born-Oppenheimer approximation Literature S.E. H Φ = E Φ H M nuclei and N electrons

Contents The Born-Oppenheimer approximation Literature S.E. H Φ = E Φ H M nuclei and N electrons Hamiltonian: H = + N i=1 N i=1 1 2 i N j>i M A=1 1 r ij + 1 2M A A M M A=1 B>A Z A Z B R AB N M i=1 A=1 Z A r ia A question Can you identify the parts of this Hamiltonian?

Contents The Born-Oppenheimer approximation Literature Conversion of atomic units to SI units

Contents The Born-Oppenheimer approximation Literature H = N i=1 1 2 i M A=1 1 2M A A N M i=1 A=1 Z A r ia + N i=1 N j>i 1 r ij + M M A=1 B>A Z A Z B R AB

Contents The Born-Oppenheimer approximation Literature N 1 M H = 2 i i=1 1 N M Z A A + 2M A r ia A=1 i=1 A=1 }{{} m N m e 0 N i=1 N j>i 1 M M Z A Z B + r ij R AB A=1 B>A }{{} m N m e const.

Contents The Born-Oppenheimer approximation Literature N 1 M H = 2 i 1 N M Z A A + i=1 2M A r ia A=1 i=1 A=1 }{{} m N m e 0 H elec = N i=1 1 2 i N M i=1 A=1 N i=1 Z A r ia + N j>i N i=1 1 M M Z A Z B + r ij R A=1 B>A AB }{{} m N m e const. N 1 j>i r ij

Contents The Born-Oppenheimer approximation Literature N 1 M H = 2 i i=1 1 N M Z A A + 2M A r ia A=1 i=1 A=1 }{{} m N m e 0 H elec = N i=1 1 2 i N M i=1 A=1 N i=1 Z A r ia + N j>i N i=1 1 M M + Z A Z B r ij R A=1 B>A AB }{{} m N m e const. N 1 j>i r ij H H elec Born-Oppenheimer approximation

Contents The Born-Oppenheimer approximation Literature H elec = N i=1 1 2 i N M i=1 A=1 Z A r ia + N i=1 N 1 j>i r ij H elec Φ elec = E elec Φ elec { }) Φ elec = Φ elec ({ r i } ; RA E elec = E elec ({ RA }) E tot = E elec + M M A=1 B>A Z A Z B R AB electronic problem

Contents The Born-Oppenheimer approximation Literature After one solves the elec. problem: H nucl = M A=1 1 ({ }) A + E RA tot 2M A

Contents The Born-Oppenheimer approximation Literature After one solves the elec. problem: H nucl = M A=1 1 ({ }) A + E RA tot 2M A }{{} potential for nuclei

Contents The Born-Oppenheimer approximation Literature Contents 1 The Born-Oppenheimer approximation 2 3 Literature

Contents The Born-Oppenheimer approximation Literature Many-electron wave functions First, let us assume non-interacting electron system... Questions 1 What will the Hamiltonian look like? 2 What will the N-electron w.f. look like? 3 What will the eigenenergies be?

Contents The Born-Oppenheimer approximation Literature Many-electron wave functions First, let us assume non-interacting electron system... H = i=1 Ψ HP ( x 1,..., x N ) = χ i ( x 1) χ k ( x N ) HΨ HP N h(i), h(i) = 1 M 2 Z A i r ia = EΨ HP E = ɛ i + + ɛ k h(i)χ j ( x i ) = ɛ j χ j ( x i ) A=1

Contents The Born-Oppenheimer approximation Literature Many-electron wave functions First, let us assume non-interacting electron system... H = N h(i), h(i) = 1 M 2 Z A i r ia i=1 A=1 Ψ HP ( x 1,..., x N ) = χ i ( x 1) χ k ( x N ) Hartree product HΨ HP = EΨ HP E = ɛ i + + ɛ k h(i)χ j ( x i ) = ɛ j χ j ( x i )

Contents The Born-Oppenheimer approximation Literature Many-electron wave functions First, let us assume non-interacting electron system... H = N h(i), h(i) = 1 M 2 Z A i r ia i=1 A=1 Ψ HP ( x 1,..., x N ) = χ i ( x 1 ) χ k ( x N ) Hartree product HΨ HP = EΨ HP E = ɛ i + + ɛ k h(i)χ j ( x i ) = ɛ j χ j ( x i ) A question Can you remember what are the problems with Ψ HP?

Contents The Born-Oppenheimer approximation Literature Many-electron wave functions First, let us assume non-interacting electron system... H = N h(i), h(i) = 1 M 2 Z A i r ia i=1 A=1 Ψ HP ( x 1,..., x N ) = χ i ( x 1 ) χ k ( x N ) Hartree product HΨ HP = EΨ HP E = ɛ i + + ɛ k h(i)χ j ( x i ) = ɛ j χ j ( x i ) A question Can you remember what are the problems with Ψ HP? 1 uncorrelated w.f. 2 doesn t account for the indistinguishability of electrons

Contents The Born-Oppenheimer approximation Literature Many-electron wave functions First, let us assume non-interacting electron system... H = N h(i), h(i) = 1 M 2 Z A i r ia i=1 A=1 Ψ HP ( x 1,..., x N ) = χ i ( x 1 ) χ k ( x N ) Hartree product HΨ HP = EΨ HP E = ɛ i + + ɛ k h(i)χ j ( x i ) = ɛ j χ j ( x i ) A question Can you remember what are the problems with Ψ HP? 1 uncorrelated w.f. 2 doesn t account for the indistinguishability of electrons So, what kind of w.f. we need?

Contents The Born-Oppenheimer approximation Literature Many-electron wave functions Slater determinant χ i ( x 1) χ j ( x 1) χ k ( x 1) Ψ( x 1,..., x N ) = 1 χ i ( x 2) χ j ( x 2) χ k ( x 2) N... χ i ( x N ) χ j ( x N ) χ k ( x N ) or shortly Ψ( x 1,..., x N ) = χ i χ j χ k Questions 1 How can you easily verify that Slater determinant satisfies the antisymmetry principle? 2 Do you know what exchange and correlation effects are? 3 Do you now what it Fermi hole is?

Contents The Born-Oppenheimer approximation Literature D. R. Hartree (1927) - term SCF, later Hartree method J. C. Slater and J. A. Gaunt (1928) - applied the variational principle J. C. Slater and V. A. Fock (1930) - used the Slater determinants

Contents The Born-Oppenheimer approximation Literature Remember the variational principle? E 0 = Ψ 0 H Ψ 0, Ψ 0 = χ i χ j χ k

Contents The Born-Oppenheimer approximation Literature Remember the variational principle? E 0 = Ψ 0 H Ψ 0, Ψ 0 = χ i χ j χ k The Hartree-Fock equation = f (i)χ( x i ) = ɛχ( x i ) Fock operator f (i) = 1 2 i M A=1 Z A r ia + v HF (i) You can learn the derivation from Ref. [1]. A question Can you identify the terms in the Fock operator?

Contents The Born-Oppenheimer approximation Literature The Hartree-Fock equation = f (i)χ( x i ) = ɛχ( x i ) Fock operator f (i) = 1 2 i M A=1 Z A r ia + v HF (i) Hartree-Fock approximation va HF (1) = χ b (2) 2 r 1 12 d x 2 - average potential b a N-elec. problem 1-elec. problem A question Can you see the problem here?

Contents The Born-Oppenheimer approximation Literature Hartree-Fock equation is nonlinear SCF (self-consistent field method)

Contents The Born-Oppenheimer approximation Literature What do we get from HF problem: {χ k } - HF spin orbitals {ɛ k } - orbital energies

Contents The Born-Oppenheimer approximation Literature What do we get from HF problem: {χ k } - HF spin orbitals {ɛ k } - orbital energies Ψ 0 = χ 1 χ aχ b χ N HF ground state w.f. A question What do you think, how many solutions of the Hf equation are there?

Contents The Born-Oppenheimer approximation Literature So, the basic strategy is: 1 choose a finite spatial basis {φ µ( r) µ = 1, 2,..., K} K 2 expand ψ i ( r) = c µφ µ( r) µ=1 3 obtain matrix equations for c µ Roothaan equations 4 get 2K total = K α + K β spin orbitals 5 2K = N occ + (2K N) unocc

Contents The Born-Oppenheimer approximation Literature So, the basic strategy is: 1 choose a finite spatial basis {φ µ( r) µ = 1, 2,..., K} K 2 expand ψ i ( r) = c µφ µ( r) µ=1 3 obtain matrix equations for c µ Roothaan equations 4 get 2K total = K α + K β spin orbitals 5 2K = N occ + (2K N) unocc The moral of the story... {φ µ} larger E 0 lower But there is a limit Hartree-Fock limit

Contents The Born-Oppenheimer approximation Literature The minimal basis H 2 model MO-LCAO = Molecular Orbitals as Linear Combinations of Atomic Orbitals atom 1 φ 1( r) atom 2 φ 2( r)

Contents The Born-Oppenheimer approximation Literature The minimal basis H 2 model MO-LCAO = Molecular Orbitals as Linear Combinations of Atomic Orbitals atom 1 φ 1( r) atom 2 φ 2( r) Slater orbitals ( ) φ( r R) ζ 3 1/2 = e ζ r R π Gaussian orbitals φ( r R) = ( ) 3/4 2α e α r R 2 π

Contents The Born-Oppenheimer approximation Literature The minimal basis H 2 model What are the properties of φ? normalized not orthogonal = S 12 = φ 1( r) φ 2( r) 0 A question What is the value of S 12 when R 12 = 0 or R 12 =?

Contents The Born-Oppenheimer approximation Literature The minimal basis H 2 model What are the properties of φ? normalized not orthogonal = S 12 = φ 1( r) φ 2( r) 0 MO-LCAO: 1 symmetric LC ψ 1 = [2(1 + S 12)] 2 1 (φ1 + φ 2) 2 antisymmetric LC ψ 2 = [2(1 S 12)] 2 1 (φ1 φ 2)

Contents The Born-Oppenheimer approximation Literature The minimal basis H 2 model What are the properties of φ? normalized not orthogonal = S 12 = φ 1( r) φ 2( r) 0 MO-LCAO: 1 symmetric LC ψ 1 = [2(1 + S 12)] 2 1 (φ1 + φ 2) 2 antisymmetric LC ψ 2 = [2(1 S 12)] 2 1 (φ1 φ 2)

Contents The Born-Oppenheimer approximation Literature The minimal basis H 2 model Questions Generally speaking, how many LCs are there? Which linear combination would give the exact molecular orbitals of H 2?

Contents The Born-Oppenheimer approximation Literature The minimal basis H 2 model Questions Generally speaking, how many LCs are there? Which linear combination would give the exact molecular orbitals of H 2? φ 1 φ 2 } minimal basis model ψ i χ i A question How many spin orbitals can we form from two spatial orbitals?

Contents The Born-Oppenheimer approximation Literature The minimal basis H 2 model φ 1 φ 2 } minimal basis model ψ i χ i χ 1( x) = ψ 1( r)α(ω) χ 2( x) = ψ 1( r)β(ω) χ 3( x) = ψ 2( r)α(ω) χ 4( x) = ψ 2( r)β(ω) A question Can you give any conclusions on the orbital energies (ɛ 1,..., ɛ 4) without solving the HF equation?

Contents The Born-Oppenheimer approximation Literature The minimal basis H 2 model χ 1( x) = ψ 1( r)α(ω) χ 2( x) = ψ 1( r)β(ω) χ 3( x) = ψ 2( r)α(ω) χ 4( x) = ψ 2( r)β(ω) Hartree-Fock ground state in min. basis model: Ψ 0 = χ 1χ 2 = ψ 1ψ 1 = 11

Contents The Born-Oppenheimer approximation Literature Excited determinants HF g.s. Ψ 0 = χ 1 χ aχ b χ N A question How many determinants can be formed from the 2K > N spin orbitals?

Contents The Born-Oppenheimer approximation Literature Excited determinants HF g.s. Ψ 0 = χ 1 χ aχ b χ N A question How many determinants can be formed ( ) 2K from the 2K > N spin orbitals? N

Contents The Born-Oppenheimer approximation Literature Excited determinants HF g.s. Ψ 0 = χ 1 χ aχ b χ N singly excited determinants (χ a χ r ): Ψ r a = χ 1 χ r χ b χ N

Contents The Born-Oppenheimer approximation Literature Excited determinants HF g.s. Ψ 0 = χ 1 χ aχ b χ N singly excited determinants (χ a χ r ): Ψ r a = χ 1 χ r χ b χ N doubly excited determinants (χ a χ r, χ b χ s): Ψ rs ab = χ 1 χ r χ s χ N A question What do you think about the purpose of excited determinants?

Contents The Born-Oppenheimer approximation Literature Configuration interaction Suppose {χ i (x)} complete set = Φ(x 1) = i a i χ i (x 1) Φ(x 1, x 2) = i< j 2 1/2 b ij χ i χ j and so on...

Contents The Born-Oppenheimer approximation Literature Configuration interaction Suppose {χ i (x)} complete set = Φ(x 1) = i a i χ i (x 1) Φ(x 1, x 2) = i< j 2 1/2 b ij χ i χ j and so on... Generally, Φ(x1,..., x N ) = c Ψ0 0 + ca r Ψ r a + ra a<b r<s cab rs So, { Ψ i } = { Ψ 0, Ψ r a, Ψ rs ab,...} complete set Ψ rs ab + a<b<c r<s<t cabc rst Ψ rst abc +

Contents The Born-Oppenheimer approximation Literature Configuration interaction Configuration interaction Generally, Φ(x 1,..., x N ) = c 0 Ψ 0 + ca r Ψ r a + ra a<b r<s cab rs Ψ rs ab So, { Ψ i } = { Ψ 0, Ψ r a, Ψ rs ab,...} complete set + a<b<c r<s<t cabc rst Ψ rst abc +

Contents The Born-Oppenheimer approximation Literature Configuration interaction Configuration interaction Generally, Φ(x 1,..., x N ) = c 0 Ψ 0 + ca r Ψ r a + ra a<b r<s cab rs Ψ rs ab So, { Ψ i } = { Ψ 0, Ψ r a, Ψ rs ab,...} complete set + a<b<c r<s<t cabc rst Ψ rst abc + Correlation energy E corr = E 0 E 0 E 0 = Ψ i H Ψj exact g.s. energy E 0 = Ψ 0 H Ψ0 HF limit energy

Contents The Born-Oppenheimer approximation Literature Configuration interaction Configuration interaction Generally, Φ(x1,..., x N ) = c Ψ0 0 + ca r Ψ r a + ra a<b r<s cab rs So, { Ψ i } = { Ψ 0, Ψ r a, Ψ rs ab,...} complete set Ψ rs ab + a<b<c r<s<t cabc rst Ψ rst abc + A question Can we reach (practically) exact eigenstates and eigenenergies?

Contents The Born-Oppenheimer approximation Literature Configuration interaction Configuration interaction Generally, Φ(x 1,..., x N ) = c 0 Ψ 0 + ca r Ψ r a + ra a<b r<s cab rs Ψ rs ab So, { Ψ i } = { Ψ 0, Ψ r a, Ψ rs ab,...} complete set + a<b<c r<s<t cabc rst Ψ rst abc + A question Can we reach (practically) exact eigenstates and eigenenergies? A solution - full CI Work with a finite set {χ i i = 1, 2,..., 2K}. Pros Exact within the subspace spanned by them. Cons Not a complete N-electron basis.

Contents The Born-Oppenheimer approximation Literature Configuration interaction

Contents The Born-Oppenheimer approximation Literature Configuration interaction Let us try CI with minimal basis H 2 model... ( ) 2K K = 2 2K = 4 = = 6 N

Contents The Born-Oppenheimer approximation Literature Configuration interaction HF g.s. Ψ0 = 11

Contents The Born-Oppenheimer approximation Literature Configuration interaction HF g.s. Ψ0 = 11 Singly excited dets Ψ 2 1 Ψ 2 1 Ψ 2 1 Ψ 2 1 = 21 = 21 = 12 = 12

Contents The Born-Oppenheimer approximation Literature Configuration interaction HF g.s. Ψ 0 = 11 Singly excited dets Ψ 2 1 Ψ 2 1 Ψ 2 1 Ψ 2 1 = 21 = 21 = 12 = 12 Doubly excited dets Ψ 22 11 = 22

Contents The Born-Oppenheimer approximation Literature Configuration interaction { Ψ0, Ψ 2 1, Ψ 2 1, Ψ 2 1, Ψ 2 1, Ψ 22 11 } : complete basis for the subspace spanned by the minimal basis any w.f. (within this subspace) can be a LC of these 6 dets for example, exact g.s. w.f. of min. basis Φ0 Questions 1 Will all these 6 dets come into the LC of Φ0? 2 How will you calculate the exact energies?

Contents The Born-Oppenheimer approximation Literature Configuration interaction Questions 1 Will all these 6 dets come into the LC of Φ0? 2 How will you calculate the exact energies? 1 No. Ψ 0 symmetric Φ 0 symmetric = Φ 0 = c0 Ψ 0 + c 22 11 Ψ 22 11 2 Diagonalize the full CI matrix: Ψ0 H Ψ0 Ψ0 H Ψ 22 11 H = Ψ 22 H Ψ 0 Ψ 22 H Ψ 22 11 11 11

Contents The Born-Oppenheimer approximation Literature Orbital energies Let us repeat ɛ j - orbital energies: f χ j = ɛ j χ j, j = 1, 2,..., j = 1, 2,..., N occupied ɛ a, ɛ b j = N + 1,..., unoccupied ɛ r, ɛ s A question What would be the physical significance of ɛ j?

Contents The Born-Oppenheimer approximation Literature Orbital energies ɛ a = χ a ( x n)h( r n)χ a( x n)d x n + b a ɛ r = χ a ( x n)χ b( x k )r 1 12 χ b ( x n)χ a( x k )d x nd x k χ r ( x n)h( r n)χ r ( x n)d x n + b χ r ( x n)χ b( x k )r 1 12 χ b ( x n)χ r ( x k )d x nd x k χ a ( x n)χ b( x k )r 1 12 χ a( x n)χ b ( x k )d x nd x k χ r ( x n)χ b( x k )r 1 12 χ r ( x n)χ b ( x k )d x nd x k where h( r n) = 1 2 n A Z A r na core-hamiltonian A question Can you identify the terms in ɛ a and ɛ r?

Contents The Born-Oppenheimer approximation Literature Orbital energies ɛ a = χ a ( x n)h( r n)χ a( x n)d x n + ( b a χ a ( x n)χ b( x k )r 1 12 χ b ( x n)χ a( x k )d x nd x k ) ɛ r = χ r ( x n)h( r n)χ r ( x n)d x n + b ( χ r ( x n)χ b( x k )r 1 12 χ b ( x n)χ r ( x k )d x nd x k ) χ a ( x n)χ b( x k )r 1 12 χ a( x n)χ b ( x k )d x nd x k χ r ( x n)χ b( x k )r 1 12 χ r ( x n)χ b ( x k )d x nd x k kinetic energy plus n-e coulomb attraction energy

Contents The Born-Oppenheimer approximation Literature Orbital energies ɛ a = χ a ( x n)h( r n)χ a( x n)d x n + ( b a χ a ( x n)χ b( x k )r 1 12 χ b ( x n)χ a( x k )d x nd x k ) ɛ r = χ r ( x n)h( r n)χ r ( x n)d x n + b ( χ r ( x n)χ b( x k )r 1 12 χ b ( x n)χ r ( x k )d x nd x k ) χ a ( x n)χ b( x k )r 1 12 χ a( x n)χ b ( x k )d x nd x k χ r ( x n)χ b( x k )r 1 12 χ r ( x n)χ b ( x k )d x nd x k e-e coulomb repulsion energy

Contents The Born-Oppenheimer approximation Literature Orbital energies ɛ a = χ a ( x n)h( r n)χ a( x n)d x n + ( b a χ a ( x n)χ b( x k )r 1 12 χ b ( x n)χ a( x k )d x nd x k ) ɛ r = χ r ( x n)h( r n)χ r ( x n)d x n + b ( χ r ( x n)χ b( x k )r 1 12 χ b ( x n)χ r ( x k )d x nd x k ) χ a ( x n)χ b( x k )r 1 12 χ a( x n)χ b ( x k )d x nd x k χ r ( x n)χ b( x k )r 1 12 χ r ( x n)χ b ( x k )d x nd x k e-e exchange energy (zero for anitparallel spins)

Contents The Born-Oppenheimer approximation Literature Orbital energies ɛ a = χ a ( x n)h( r n)χ a( x n)d x n + ( b a χ a ( x n)χ b( x k )r 1 12 χ b ( x n)χ a( x k )d x nd x k ) ɛ r = χ r ( x n)h( r n)χ r ( x n)d x n + b ( χ r ( x n)χ b( x k )r 1 12 χ b ( x n)χ r ( x k )d x nd x k ) χ a ( x n)χ b( x k )r 1 12 χ a( x n)χ b ( x k )d x nd x k χ r ( x n)χ b( x k )r 1 12 χ r ( x n)χ b ( x k )d x nd x k coulomb and exchange interaction with all N electrons (as if an electron has been added to the HF g.s. Ψ0 ) A question Can you guess the relationship between ɛ a and E 0?

Contents The Born-Oppenheimer approximation Literature Orbital energies = N ɛ a = a E 0 = N ɛ a E 0 a N N N a h a + ab ab a N a h a + 1 2 a a b N N ab ab a b Why? (Hint: check the sum indeces.)

Contents The Born-Oppenheimer approximation Literature Orbital energies Imagine that we remove an electron from χ c : N Ψ 0 N 1 Ψ c = χ1 χ c 1χ c+1 χ N A question Can you figure out what energy is required for this process?

Contents The Born-Oppenheimer approximation Literature Orbital energies Imagine that we remove an electron from χ c : N Ψ 0 N 1 Ψ c = χ1 χ c 1χ c+1 χ N IP = N 1 E c N E 0 ionization potential IP = ɛ c

Contents The Born-Oppenheimer approximation Literature Orbital energies Now, imagine that we add an electron to χ r : N Ψ 0 N+1 Ψ r = χ r χ 1 χ N A question Can you figure out what energy is released during this process?

Contents The Born-Oppenheimer approximation Literature Orbital energies Now, imagine that we add an electron to χ r : N Ψ 0 N+1 Ψ r = χ r χ 1 χ N EA = N E 0 N+1 E r electron affinity EA = ɛ r

Contents The Born-Oppenheimer approximation Literature Orbital energies Koopmans theorem Given N Ψ 0 and ɛc and ɛ r, then IP = ɛ c EA = ɛ r

Contents The Born-Oppenheimer approximation Literature Contents 1 The Born-Oppenheimer approximation 2 3 Literature

Contents The Born-Oppenheimer approximation Literature Literature 1 A. Szabo, N. Ostlund, Modern Quantum Chemistry, Introduction to Advanced Electronic Structure theory, Dover Publications, New York, 1996. 2 I. Supek, Teorijska fizika i struktura materije, II. dio, Školska knjiga, Zagreb, 1989. 3 R. L. Liboff, Introductory Quantum Mechanics, Addison Wesley, San Francisco, 2003. 4 WebMO 5 List of quantum chemistry and solid-state physics software