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Second Quantization
APPENDIX C
Second Quantization When we work with a basis set composed of the Slater determinants, we are usually confronted with a large number of the matrix elements involving one- and two-electron operators. The Slater–Condon rules (Appendix V1-N) are doing the job to express these matrix elements by the one-electron and two-electron integrals. However, we may introduce an even easier tool, called second quantization, which is equivalent to the Slater–Condon rules.
Vacuum state In the second quantization formalism we introduce for the system under study a reference state, which is a Slater determinant (usually the Hartree–Fock wave function) composed of N orthonormal spin orbitals, where N is the number of electrons. This function will be denoted in short by 0 or as a particular case of N (n1 , n2 , ..., n∞ ), where the last notation means that we have to do with a normalized N-electron Slater determinant, and in the parentheses we give the occupancy list (ni = 0, 1) for the infinite number of the orthonormal spin orbitals considered in the basis set and listed one by one in the parentheses. This simply means that some spin orbitals are present in the determinant (they have ni = 1), while some other are absent1 (ni = 0). Hence, i ni = N. The reference state is often called the vacuum state. The subscript 0 in 0 means that we consider this single-determinant as an approximation to the ground state wave function. Besides the reference state some normalized Slater determinants of the excited states will be considered, with other occupancies, including those corresponding to the number of electrons that differs from N.
1 For example, the symbol 2 (001000100000...) means a normalized Slater determinant of dimension 2, containing the spin orbitals 3 and 7. The symbol 2 (001000...) makes no sense, because the number of “ones” has to
be equal to 2, etc.
587
588 Appendix C
Creation and annihilation of electrons Let us make a quite strange move and consider operators that change the number of electrons in the system. To this end let us define the creation operator2 kˆ † of the electron going to occupy the spin orbital k and the annihilation operator kˆ of an electron disappearing from the spin orbital k: CREATION AND ANNIHILATION OPERATORS kˆ † N (...nk ...) = θk (1 − nk ) N+1 (...1k , ...) , ˆ N (...nk ...) = θk nk N−1 (...0k , ...) , k where θk = (−1)j
Richard Feynman in one of his books says jokingly that he could not understand the very sense of the operators. If we annihilate or create an electron, then what about the system’s electroneutrality? Happily enough these operators will always act in creator–annihilator pairs. 3 Proof. Let us take two Slater determinants = N+1 (...1 ...) and = N (...0 ...). In both of them the a k b k occupancies Let us of all other spinorbitals are identical. write the normalization condition for b in the following ˆ a = θk kˆ # b |a , where as kˆ # has been denoted the operator adjoint to k, ˆ ˆ a = θk b |k way: 1 = b |θk k θk appeared in order to compensate (θk2 = 1) the θk produced by the annihilator. On the other hand, from the normalization condition of a we see that 1 = a |a = θk kˆ † b |a . Hence, θk kˆ # b |a = θk kˆ † b |a or kˆ # = kˆ † , quod erat demonstrandum.
Second Quantization 589 ANTICOMMUTATION RULES ˆ l] ˆ + = 0, [k, [kˆ † , lˆ† ]+ = 0, ˆ + = δkl , [kˆ † , l] ˆ B] ˆ + = Aˆ Bˆ + Bˆ Aˆ is called the anticommutator.4 It is simpler than the where the symbol [A, ˆ + = δkl . We have to check how it Slater–Condon rules, isn’t it? Let us check the rule [kˆ † , l] works for all possible occupancies of the spin orbitals k and l, (nk , nl ) : (0, 0), (0, 1), (1, 0), and (1, 1).
Case: (nk , nl ) = (0, 0) We have ˆ + N (...0k ...0l ...) = [kˆ † lˆ + lˆkˆ † ]N (...0k ...0l ...) = [kˆ † , l] ˆ N (...0k ...0l ...) + lˆkˆ † N (...0k ...0l ...) = kˆ † l ˆ N+1 (...1k ...0l ...) = θk δkl θk N (...0k ...) = δkl N (...0k ...). ˆ k N+1 (...1k ...0l ...) = θk l 0 + lθ So far so good.
Case: (nk , nl ) = (0, 1) We have ˆ + N (...0k ...1l ...) = [kˆ † lˆ + lˆkˆ † ]N (...0k ...1l ...) = [kˆ † , l] ˆ N (...0k ...1l ...) + lˆkˆ † N (...0k ...1l ...) = kˆ † l θk θl N (...1k ...0l ...) − θk θl N (...1k ...0l ...) = δkl N (...0k ...1l ...). That is what we expected.5 4 The above formulae are valid under the (common) assumption that the spin orbitals are orthonormal. If this ˆ + = Skl , where Skl stands assumption is not true, then only the last anticommutator changes to the form [kˆ † , l]
for the overlap integral of spin orbitals k and l. 5 What decided is the change of sign (due to θ ) when the order of the operators has changed. k
590 Appendix C
Case: (nk , nl ) = (1, 0) We have ˆ + N (...1k ...0l ...) = [kˆ † lˆ + lˆkˆ † ]N (...1k ...0l ...) = [kˆ † , l] ˆ N (...1k ...0l ...) + lˆkˆ † N (...1k ...0l ...) = kˆ † l (0 + 0) N (...1k ...0l ...) = δkl N (...1k ...0l ...). That is OK.
Case: (nk , nl ) = (1, 1) We have ˆ + N (...1k ...1l ...) = [kˆ † lˆ + lˆkˆ † ]N (...1k ...1l ...) = [kˆ † , l] ˆ N (...1k ...1l ...) + lˆkˆ † N (...1k ...1l ...) = kˆ † l ˆ N (...1k ...1l ...) + 0 = θk2 δkl N (...1k ...1l ...) = δkl N (...1k ...). kˆ † l OK, the thing is over.
Operators in the second quantization The creation and annihilation operators may be used to represent the one- and two-electron operators. The resulting matrix elements with Slater determinants6 correspond exactly to the Slater–Condon rules (see Appendix V1-N, p. V1-707).
One-electron operators The operator Fˆ = i hˆ (i) is a sum of the one-electron operators7 hˆ (i) acting on functions of the coordinates of electron i. Slater–Condon rule I says (see Appendix V1-N) ψ built of the that for the Slaterdeterminant ˆ j . spin orbitals φi the matrix element ψ|Fˆ ψ = i hii , where hij = φi |hφ 6 The original operator and its representation in the language of the second quantization are not identical though.
The second ones can act only on the Slater determinants or their combinations, while the first ones have a larger domain. Since we are going to work with the creation and annihilation operators in those methods only that use Slater determinants (CI, MC SCF, etc.), the difference is irrelevant. 7 Most often this will be the kinetic energy operator, the nuclear attraction operator, the interaction with the external field, or the multipole moment.
Second Quantization 591 In the second quantization Fˆ =
∞
hij iˆ† jˆ.
ij
Interestingly, the summation extends to infinity, and, therefore, the operator is independent of the number of electrons in the system. Let us check whether the formula is correct. Let us insert Fˆ = ij hij ıˆ† jˆ into ψ|Fˆ ψ . We have ψ|Fˆ ψ = ψ| hij ıˆ† jˆψ = hij ψ|ˆı † jˆψ = hij δij = hii . ij
ij
ij
i
This is a correct result. What about Slater–Condon rule II (the Slater determinants ψ1 and ψ2 differ by a single spin orbital: the spin orbital i in ψ1 is replaced by the spin orbital i in ψ2 )? We have ψ1 |Fˆ ψ2 = hij ψ1 |ˆı † jˆψ2 . ij
The Slater determinants by one spin orbital produce the overlap integral equal to that differ 8 zero, and therefore ψ1 |Fˆ ψ2 = hii . Thus, the operator in the form Fˆ = ij hij ıˆ† jˆ ensures equivalence with all the Slater–Condon rules.
Two-electron operators Similarly, we may use the creation and annihilation operators to represent the two-electron ˆ = 1 ij gˆ (i, j ). In most cases gˆ (i, j ) = 1 and G ˆ has the following form: operators G 2 rij
∞
1 1 ˆ=1 ˆ ij |kl jˆ† iˆ† kˆ l. G = 2 rij 2 ij
ij kl
8 It is evident that if in this situation the Slater determinants ψ and ψ differed by more than a single spin orbital, 1 2
then we would get zero (Slater–Condon rules III and IV).
592 Appendix C Here also the summation extends to infinity and the operator is independent of the number of electrons in the system. The proof of Slater–Condon rule I relies on the following chain of equalities: 1 ˆ =1 ˆ = ˆ ij |kl ψ|jˆ† ıˆ† kˆ lψ ij |kl ıˆjˆψ|kˆ lψ ψ|Gψ = 2 2 1 2
ij kl
ij |kl δik δj l − δil δj k =
ij kl
1 2
ij kl
(ij |ij − ij |j i) ,
ij
ˆ ˆ ˆ is nonzero because the overlap integral ıˆjˆψ|k lψ of the two Slater determinants ıˆjˆψ and kˆ lψ in two cases only: either if i = k, j = l or if i = l, j = k (then the sign has to change). This is what we get from the Slater–Condon rules. For Slater–Condon rule II we have (instead of the spin orbital i in ψ1 we have the spin orbital i in ψ2 ) ˆ 2 =1 ˆ 2 =1 ˆ 2 , Ij |kl ψ1 |jˆ† Iˆ† kˆ lψ Ij |kl Iˆjˆψ1 |kˆ lψ (C.1) ψ1 |Gψ 2 2 Ij kl
Ij kl
where the summation index I has been introduced in order not to mix up with spin orbital i. ˆ 2 the sets of the spin orbitals in the Slater determinant Iˆjˆψ1 In the overlap integral Iˆjˆψ1 |kˆ lψ ˆ 2 have to be identical; otherwise the integral will be equal to and in the Slater determinant kˆ lψ zero. However, already in ψ1 and ψ2 we have a difference of one spin orbital. Thus, first of all we have to get rid of just these spin orbitals (i and i )! For the integral to survive9 we have to have at least one of the following conditions to be satisfied: • • • •
I j I j
=i =i =i =i
and k = i (and then j = l), and k = i (and then I = l), and l = i (and then j = k), and l = i (and then I = k).
This means that when taking into account the above cases in Eq. (C.1) we obtain
1
ˆı ψ1 |ˆı lψ ˆ 2 ˆ 2 =1 ij |i j ıˆjˆψ1 |ˆı jˆψ2 + li|i l lˆ ψ1 |Gψ 2 2 j l
1 1 1 ˆ ı ψ1 |kˆ ˆ ı ψ2 = 1 ij |j i ıˆjˆψ1 |jˆıˆ ψ2 + ki|ki kˆ ij |i j − li|i l , + 2 2 2 2 j
k
9 This is necessary, not a sufficient condition.
j
l
Second Quantization 593 −
1
1 1
1
ij |j i + ki|ki = ij |i j − j i|i j , 2 2 2 2 j
−
1
2
k
ij |j i +
j
1
2
j
j i|j i
j
=
1
2
j
j
ij |i j −
1
2
ij |j i ,
j
1
1
− ij |j i + ij |i j = ij |i j − ij |j i , 2 2 j
j
j
j
where in the two sums the coordinates of electrons 1 and 2 have been exchanged, and it has
ˆ ı ψ1 |kˆ ˆ ı ψ2 = 1, because the Slater debeen noted that the overlap integrals ıˆjˆψ1 |ˆı jˆψ2 = kˆ ˆı ψ1 |ˆı lψ ˆ 2 = ˆ 1 and iˆ ψ2 are identical. Also, from the anticommutation rules lˆ terminants iψ
ˆ 2 = ıˆjˆψ1 |jˆıˆ ψ2 = −1. Thus, Slater–Condon rule II has been correctly reproduced: ψ1 |Gψ
j ij |i j − ij |j i . We may conclude that the definition of the creation and annihilation operators and the simple anticommutation relations are equivalent to the Slater–Condon rules. This opens up for us the space spanned by the Slater determinants, i.e., all the integrals involving Slater determinants can be easily transformed into the one- and two-electron integrals involving spin orbitals.
Second Quantization When we work with a basis set composed of the Slater determinants, we are usually confronted with a large number of the matrix elements involving one- and two-electron operators. The Slater–Condon rules (Appendix V1-N) are doing the job to express these matrix elements by the one-electron and two-electron integrals. However, we may introduce an even easier tool, called second quantization, which is equivalent to the Slater–Condon rules.
Vacuum state In the second quantization formalism we introduce for the system under study a reference state, which is a Slater determinant (usually the Hartree–Fock wave function) composed of N orthonormal spin orbitals, where N is the number of electrons. This function will be denoted in short by 0 or as a particular case of N (n1 , n2 , ..., n∞ ), where the last notation means that we have to do with a normalized N-electron Slater determinant, and in the parentheses we give the occupancy list (ni = 0, 1) for the infinite number of the orthonormal spin orbitals considered in the basis set and listed one by one in the parentheses. This simply means that some spin orbitals are present in the determinant (they have ni = 1), while some other are absent1 (ni = 0). Hence, i ni = N. The reference state is often called the vacuum state. The subscript 0 in 0 means that we consider this single-determinant as an approximation to the ground state wave function. Besides the reference state some normalized Slater determinants of the excited states will be considered, with other occupancies, including those corresponding to the number of electrons that differs from N.
1 For example, the symbol 2 (001000100000...) means a normalized Slater determinant of dimension 2, containing the spin orbitals 3 and 7. The symbol 2 (001000...) makes no sense, because the number of “ones” has to
be equal to 2, etc.
587
588 Appendix C
Creation and annihilation of electrons Let us make a quite strange move and consider operators that change the number of electrons in the system. To this end let us define the creation operator2 kˆ † of the electron going to occupy the spin orbital k and the annihilation operator kˆ of an electron disappearing from the spin orbital k: CREATION AND ANNIHILATION OPERATORS kˆ † N (...nk ...) = θk (1 − nk ) N+1 (...1k , ...) , ˆ N (...nk ...) = θk nk N−1 (...0k , ...) , k where θk = (−1)j
Richard Feynman in one of his books says jokingly that he could not understand the very sense of the operators. If we annihilate or create an electron, then what about the system’s electroneutrality? Happily enough these operators will always act in creator–annihilator pairs. 3 Proof. Let us take two Slater determinants = N+1 (...1 ...) and = N (...0 ...). In both of them the a k b k occupancies Let us of all other spinorbitals are identical. write the normalization condition for b in the following ˆ a = θk kˆ # b |a , where as kˆ # has been denoted the operator adjoint to k, ˆ ˆ a = θk b |k way: 1 = b |θk k θk appeared in order to compensate (θk2 = 1) the θk produced by the annihilator. On the other hand, from the normalization condition of a we see that 1 = a |a = θk kˆ † b |a . Hence, θk kˆ # b |a = θk kˆ † b |a or kˆ # = kˆ † , quod erat demonstrandum.
Second Quantization 589 ANTICOMMUTATION RULES ˆ l] ˆ + = 0, [k, [kˆ † , lˆ† ]+ = 0, ˆ + = δkl , [kˆ † , l] ˆ B] ˆ + = Aˆ Bˆ + Bˆ Aˆ is called the anticommutator.4 It is simpler than the where the symbol [A, ˆ + = δkl . We have to check how it Slater–Condon rules, isn’t it? Let us check the rule [kˆ † , l] works for all possible occupancies of the spin orbitals k and l, (nk , nl ) : (0, 0), (0, 1), (1, 0), and (1, 1).
Case: (nk , nl ) = (0, 0) We have ˆ + N (...0k ...0l ...) = [kˆ † lˆ + lˆkˆ † ]N (...0k ...0l ...) = [kˆ † , l] ˆ N (...0k ...0l ...) + lˆkˆ † N (...0k ...0l ...) = kˆ † l ˆ N+1 (...1k ...0l ...) = θk δkl θk N (...0k ...) = δkl N (...0k ...). ˆ k N+1 (...1k ...0l ...) = θk l 0 + lθ So far so good.
Case: (nk , nl ) = (0, 1) We have ˆ + N (...0k ...1l ...) = [kˆ † lˆ + lˆkˆ † ]N (...0k ...1l ...) = [kˆ † , l] ˆ N (...0k ...1l ...) + lˆkˆ † N (...0k ...1l ...) = kˆ † l θk θl N (...1k ...0l ...) − θk θl N (...1k ...0l ...) = δkl N (...0k ...1l ...). That is what we expected.5 4 The above formulae are valid under the (common) assumption that the spin orbitals are orthonormal. If this ˆ + = Skl , where Skl stands assumption is not true, then only the last anticommutator changes to the form [kˆ † , l]
for the overlap integral of spin orbitals k and l. 5 What decided is the change of sign (due to θ ) when the order of the operators has changed. k
590 Appendix C
Case: (nk , nl ) = (1, 0) We have ˆ + N (...1k ...0l ...) = [kˆ † lˆ + lˆkˆ † ]N (...1k ...0l ...) = [kˆ † , l] ˆ N (...1k ...0l ...) + lˆkˆ † N (...1k ...0l ...) = kˆ † l (0 + 0) N (...1k ...0l ...) = δkl N (...1k ...0l ...). That is OK.
Case: (nk , nl ) = (1, 1) We have ˆ + N (...1k ...1l ...) = [kˆ † lˆ + lˆkˆ † ]N (...1k ...1l ...) = [kˆ † , l] ˆ N (...1k ...1l ...) + lˆkˆ † N (...1k ...1l ...) = kˆ † l ˆ N (...1k ...1l ...) + 0 = θk2 δkl N (...1k ...1l ...) = δkl N (...1k ...). kˆ † l OK, the thing is over.
Operators in the second quantization The creation and annihilation operators may be used to represent the one- and two-electron operators. The resulting matrix elements with Slater determinants6 correspond exactly to the Slater–Condon rules (see Appendix V1-N, p. V1-707).
One-electron operators The operator Fˆ = i hˆ (i) is a sum of the one-electron operators7 hˆ (i) acting on functions of the coordinates of electron i. Slater–Condon rule I says (see Appendix V1-N) ψ built of the that for the Slaterdeterminant ˆ j . spin orbitals φi the matrix element ψ|Fˆ ψ = i hii , where hij = φi |hφ 6 The original operator and its representation in the language of the second quantization are not identical though.
The second ones can act only on the Slater determinants or their combinations, while the first ones have a larger domain. Since we are going to work with the creation and annihilation operators in those methods only that use Slater determinants (CI, MC SCF, etc.), the difference is irrelevant. 7 Most often this will be the kinetic energy operator, the nuclear attraction operator, the interaction with the external field, or the multipole moment.
Second Quantization 591 In the second quantization Fˆ =
∞
hij iˆ† jˆ.
ij
Interestingly, the summation extends to infinity, and, therefore, the operator is independent of the number of electrons in the system. Let us check whether the formula is correct. Let us insert Fˆ = ij hij ıˆ† jˆ into ψ|Fˆ ψ . We have ψ|Fˆ ψ = ψ| hij ıˆ† jˆψ = hij ψ|ˆı † jˆψ = hij δij = hii . ij
ij
ij
i
This is a correct result. What about Slater–Condon rule II (the Slater determinants ψ1 and ψ2 differ by a single spin orbital: the spin orbital i in ψ1 is replaced by the spin orbital i in ψ2 )? We have ψ1 |Fˆ ψ2 = hij ψ1 |ˆı † jˆψ2 . ij
The Slater determinants by one spin orbital produce the overlap integral equal to that differ 8 zero, and therefore ψ1 |Fˆ ψ2 = hii . Thus, the operator in the form Fˆ = ij hij ıˆ† jˆ ensures equivalence with all the Slater–Condon rules.
Two-electron operators Similarly, we may use the creation and annihilation operators to represent the two-electron ˆ = 1 ij gˆ (i, j ). In most cases gˆ (i, j ) = 1 and G ˆ has the following form: operators G 2 rij
∞
1 1 ˆ=1 ˆ ij |kl jˆ† iˆ† kˆ l. G = 2 rij 2 ij
ij kl
8 It is evident that if in this situation the Slater determinants ψ and ψ differed by more than a single spin orbital, 1 2
then we would get zero (Slater–Condon rules III and IV).
592 Appendix C Here also the summation extends to infinity and the operator is independent of the number of electrons in the system. The proof of Slater–Condon rule I relies on the following chain of equalities: 1 ˆ =1 ˆ = ˆ ij |kl ψ|jˆ† ıˆ† kˆ lψ ij |kl ıˆjˆψ|kˆ lψ ψ|Gψ = 2 2 1 2
ij kl
ij |kl δik δj l − δil δj k =
ij kl
1 2
ij kl
(ij |ij − ij |j i) ,
ij
ˆ ˆ ˆ is nonzero because the overlap integral ıˆjˆψ|k lψ of the two Slater determinants ıˆjˆψ and kˆ lψ in two cases only: either if i = k, j = l or if i = l, j = k (then the sign has to change). This is what we get from the Slater–Condon rules. For Slater–Condon rule II we have (instead of the spin orbital i in ψ1 we have the spin orbital i in ψ2 ) ˆ 2 =1 ˆ 2 =1 ˆ 2 , Ij |kl ψ1 |jˆ† Iˆ† kˆ lψ Ij |kl Iˆjˆψ1 |kˆ lψ (C.1) ψ1 |Gψ 2 2 Ij kl
Ij kl
where the summation index I has been introduced in order not to mix up with spin orbital i. ˆ 2 the sets of the spin orbitals in the Slater determinant Iˆjˆψ1 In the overlap integral Iˆjˆψ1 |kˆ lψ ˆ 2 have to be identical; otherwise the integral will be equal to and in the Slater determinant kˆ lψ zero. However, already in ψ1 and ψ2 we have a difference of one spin orbital. Thus, first of all we have to get rid of just these spin orbitals (i and i )! For the integral to survive9 we have to have at least one of the following conditions to be satisfied: • • • •
I j I j
=i =i =i =i
and k = i (and then j = l), and k = i (and then I = l), and l = i (and then j = k), and l = i (and then I = k).
This means that when taking into account the above cases in Eq. (C.1) we obtain
1
ˆı ψ1 |ˆı lψ ˆ 2 ˆ 2 =1 ij |i j ıˆjˆψ1 |ˆı jˆψ2 + li|i l lˆ ψ1 |Gψ 2 2 j l
1 1 1 ˆ ı ψ1 |kˆ ˆ ı ψ2 = 1 ij |j i ıˆjˆψ1 |jˆıˆ ψ2 + ki|ki kˆ ij |i j − li|i l , + 2 2 2 2 j
k
9 This is necessary, not a sufficient condition.
j
l
Second Quantization 593 −
1
1 1
1
ij |j i + ki|ki = ij |i j − j i|i j , 2 2 2 2 j
−
1
2
k
ij |j i +
j
1
2
j
j i|j i
j
=
1
2
j
j
ij |i j −
1
2
ij |j i ,
j
1
1
− ij |j i + ij |i j = ij |i j − ij |j i , 2 2 j
j
j
j
where in the two sums the coordinates of electrons 1 and 2 have been exchanged, and it has
ˆ ı ψ1 |kˆ ˆ ı ψ2 = 1, because the Slater debeen noted that the overlap integrals ıˆjˆψ1 |ˆı jˆψ2 = kˆ ˆı ψ1 |ˆı lψ ˆ 2 = ˆ 1 and iˆ ψ2 are identical. Also, from the anticommutation rules lˆ terminants iψ
ˆ 2 = ıˆjˆψ1 |jˆıˆ ψ2 = −1. Thus, Slater–Condon rule II has been correctly reproduced: ψ1 |Gψ
j ij |i j − ij |j i . We may conclude that the definition of the creation and annihilation operators and the simple anticommutation relations are equivalent to the Slater–Condon rules. This opens up for us the space spanned by the Slater determinants, i.e., all the integrals involving Slater determinants can be easily transformed into the one- and two-electron integrals involving spin orbitals.