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HfHartree Fock

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Second quantization
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  1 Hartree-Fock Approximation We have a system of   N   interacting fermions with a Hamiltonian H   = N   i =1 h ( i ) + N   i<j V  int ( i,j )  .  (1)Here,  h ( i ) is the “single-particle Hamiltonian” for the particle- i  and  V  int ( i,j )is the part of the Hamiltonian describing the interaction between particles  i and  j . By the “single-particle Hamiltonian” we mean the terms that wouldbe present if there were only one fermion in the system. For the case of   N  electrons in an atom with proton number  Z   at the nucleus, these terms wouldbe h ( i ) =  p 2 i 2 m  −  Ze 2 r i ,V  int ( i,j ) =  e 2 | r i − r  j  |  . Again we ignored the rest of the terms like spin-orbit interaction which wouldintroduce spin-dependent terms into the Hamiltonian. Note that in Eq. (1),the interaction part is summed over each particle pair only once. As a result,there are    N  2  =  N  ( N   − 1)2terms in the last sum.Such problems cannot be solved exactly in general (there are a few unin-teresting exceptions). As a result, we need a general approximation procedurefor solving such many-body equations. Obviously, what we want to calcu-late is important in choosing the approximation procedure. Hence, thereare various methods developed to cope with different aspects of such kindof problems. The very first step should be obtaining a “feel” for the energylevels for those problems. That certainly means that we don’t need accuratenumerical values.Independent particle picture is this first step in visualizing the energylevels of most of the many-body problems. This is the language used indescribing the electronic states of atoms in Atomic Physics courses. Whenwe are saying that in Lithium, there are two electrons in the 1s orbital and1  one electron in the 2s orbital, we are using that language. However, we say, inthose atoms, the orbital wavefunctions and their energies are different thanthose of the hydrogenic atoms. Our purpose in here is to formalize thosekind of statements.Therefore, the question we would like to answer is “What is the bestindependent-particle approximation for an interacting system of fermions?”And the answer can be provided by the variational theory. We should forma many-body wavefunction of the atom as a Slater determinant using someone-particle states  ϕ 1 ,...,ϕ N  . And, we should choose these one-particlestates as the ones that minimize the expectation value of the Hamiltonian inEq. (1). This is the Hartree-Fock approximation. 2 Expectation Values Let us suppose that we have  N   arbitrary one-particle states  ϕ 1 ,...,ϕ N  . Weshould choose them as an orthonormal set, since this will make the calculationof expectation values easier. So, we have  ϕ i  | ϕ  j  =  δ  i,j  i,j  = 1 ,...,N.  (2)As a result, we have  N  2 equations giving us a restriction on the one-particlestates. (Some may notice that, there are only  N  ( N   + 1) / 2 equations inEq. (2), but the number of real quantities fixed to a certain value is  N  2 . So,there are  N  2 real equations.)We form the Slater determinant from these  N   states asΨ(1 , 2 ,...,N  ) = 1 √  N  !  ϕ 1 (1)  ϕ 1 (2)  ... ϕ 1 ( N  ) ϕ 2 (1)  ϕ 2 (2)  ... ϕ 2 ( N  )... ... ... ... ϕ N  (1)  ϕ N  (2)  ... ϕ N  ( N  )  . By using the many-body wavefunction thus formed, we will calculate theexpectation value of the Hamiltonian,  H  , as  Ψ | H   | Ψ  =  N   Ψ | h (1) | Ψ  +   N  2   Ψ | V  int (1 , 2) | Ψ   . These quantities were calculated before, so we just reproduce the results inhere  Ψ | H   | Ψ  = N   i =1 E  i  +  i<j ( C  ij − E  ij )  ,  (3)2  where E  i  =   ϕ i  | h | ϕ i   ,C  ij  =   ϕ i ϕ  j  | V  int  | ϕ i ϕ  j   ,E  ij  =   ϕ  j ϕ i  | V  int  | ϕ i ϕ  j   . Here,  C  ij  is usually called as the Coulomb integral for the orbitals  ϕ i  and ϕ  j  (specifically considering electrons in atoms and molecules), and  E  ij  iscalled the exchange integral. We note that in Eq. (3) sum over Coulomband exchange integrals is over pairs of states. That sum can be written inanother way. We just need to note that  C  ij  =  C   ji  and  E  ij  =  E   ji . It canalso be noticed that, by the definition of Coulomb and exchange integrals,we have the relation  C  ii  =  E  ii . Keep in mind that  C  ii  is not physicallymeaningful (interaction of an electron with itself?). But, in order to simplifythe Eq. (3), those terms are frequently used. By using these relations, Eq. (3)can be rewritten as,  Ψ | H   | Ψ  = N   i =1 E  i  + 12 N   i =1 N    j =1 ( C  ij − E  ij )  .  (4) 3 Effective Hamiltonian Now, all we have to do is to minimize the expectation value in Eq. (4) withthe restrictions in Eq. (2). We can use the Lagrange multiplier method tocarry out this minimization. As a result the quantity, A  =  Ψ | H   | Ψ − N   i =1 N    j =1 Λ ij  ϕ i  | ϕ  j   (5)should be minimized, where Λ ij  are the Lagrange multipliers. The Lagrangemultipliers in this expression can take on complex values, but you shouldnot be concerned about this. Obviously, only real valued functions can beminimized, hence the expression in Eq. (5) should be real valued. This impliesthat the Λ matrix is hermitian. If you are concerned about the presenceof complex numbers in a real-valued expression, you should derive Eq. (5)starting from the real-valued forms of the expressions in Eq. (2).3  Now, we can consider the variations in each element of the one-particlestate set  ϕ 1 ,...,ϕ N  . In effect, what we will be doing is evaluating the func-tional derivatives and equating them to zero. Since we are using the Lagrangemultiplier method, we can vary each  ϕ i  in an arbitrary way. That means,we will make sure that the replacement  |  ϕ i  →|  ϕ i  +  |  δϕ i  , where  |  δϕ i  is an arbitrary infinitesimal state, will not change the expression in Eq. (4)up to first order. (To be precise, we will set  |  δϕ i   =    |  ψ   where  |  ψ   isan arbitrary one-particle state and we will equate the order    term of thevariation in Eq. (4) to 0.)Now, the first order variation of real-valued expressions that depend ona quantum mechanical state (say  | ϕ i  ) are of the form  δϕ i  | α  +  α | δϕ i  for some (perhaps complicated) “state”  |  α  . This expression is then setequal to 0 at the extrema. Since  | δϕ i   is arbitrary, this will only imply that | α  = 0 (show). If we keep this in mind, we can quickly find the conditionsfor minimum in our calculation. As a result we find, δA  = N   i =1  δϕ i  | h | ϕ i  + N   i,j =1   δϕ i ϕ  j  | V  int  | ϕ i ϕ  j − δϕ i ϕ  j  | V  int  | ϕ  j ϕ i   − N   i,j =1 Λ ij  δϕ i  | ϕ  j  +  complex-conjugate  . Now, if arbitrary infinitesimal values of   |  δϕ i   makes the expression abovevanish, the the following equation should be satisfied for the  i th state:( h + V  Coul  + V  Exch ) | ϕ i  = N    j =1 Λ ij  | ϕ  j   ,  (6)where  V  Coul  and  V  Exch  are operators which are a little bit complicated. In theabstract case that we are using in here, we will define them by their matrix4
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