Here I give a part of derivation of Hartree-Fock equations in case where basis functions (wavefunctions) are orthonormal and real: $$ \langle \psi_i | \psi_j \rangle = \langle \psi_j | \psi_i \rangle = \delta_{ij} $$
Trial wavefunction is defined as: $$ |\Phi \rangle = \sum_{i=1}^n c_i |\psi_i \rangle $$
where $|\psi_i\rangle$ is basis function $i$.
Expectation value of energy is given by: $$ \langle \Phi | H |\Phi \rangle = \sum_{ij} c_i c_j \langle \psi_i |H|\psi_j \rangle $$
I don't quite understand, why is expectation value of energy for trial wavefunction equal sum of expectation values for every combination of two basis functions multiplied by their respective coefficients ($c_i$ and $c_j$)? What justifies this summation?
Best Answer
It's really linear algebra: if $$ \left| \Phi \right \rangle = \sum_{i} c_i \left| \Psi_i \right \rangle $$ taking Hermitian operator ("transpose conjugate"): $$ \left| \Phi \right \rangle^\dagger = \left\langle \Phi \right| = \sum_{i} c^*_i \left\langle \Psi_i \right|. $$ Now "sandwiching" $H$ you get: $$ \left\langle \Phi \right| H \left| \Phi \right \rangle = \left(\sum_{i} c^*_i \left\langle \Psi_i \right| \right) H \left(\sum_{i} c_i \left| \Psi_i \right \rangle \right) $$ now, before we expand, we should change one of the indices to $j$ to account for products of different terms, just like, say: $$ (a_1 + a_2) \times (b_1 + b_2) = a_1 b_1 + a_1 b_2 + a_2 b_1 + a_2 b_2 = \sum_{i,j = 1}^{2} a_i b_j $$ and not $\sum_i a_i b_i$. So, really: $$ \begin{align*} \left\langle \Phi \right| H \left| \Phi \right \rangle &= \left(\sum_{i} c^*_i \left\langle \Psi_i \right| \right) H \left(\sum_{i} c_i \left| \Psi_i \right \rangle \right) \\ & = \left(\sum_{j} c^*_j \left\langle \Psi_j \right| \right)\left(\sum_{i} c_i H\left| \Psi_i \right \rangle \right) \end{align*} $$ and apply the distribution rule of algebra: $$ \left(\sum_{j} c^*_j \left\langle \Psi_j \right| \right) \left(\sum_{i} c_i H \left| \Psi_i \right \rangle \right) = \sum_{i, j} c^*_j c_i (\left\langle \Psi_j \right|) (H\left| \Psi_i \right \rangle) = \sum_{i, j} c_i c^*_j \left\langle \Psi_i \right| H\left| \Psi_j \right \rangle $$