This post is inspired by a paper by Ali Alavi and a talk by Frank Neese at ICQC 2023. The goal of the work by Frank Neese and Ali Alavi is to be able to describe a wave function with...
In a previous post the Jordan-Wigner representation of the molecular Hamiltonian for the hydrogen molecule was constructed. This Hamiltonian was solved explicit diagonalization. The method of explicit diagonalization quickly becomes computationally impossible when the dimension of the Hamiltonian grow, and...
The first step of using quantum computers to perform quantum chemical simulations is to translate the fermionic Hamiltonian to a qubit Hamiltonian (represented by Pauli operators). The fermionic Hamiltonian in spin basis is: \[H = \sum_{pq}h_{pq}a^\dagger_p a_q + \frac{1}{2}\sum_{pqrs}g_{pqrs}a^\dagger_p a^\dagger_q...
The hydrogen molecule in a minimal basis (STO-3G as an example) will have a full CI Hamiltonian that is only four by four (using determinants that are singlet). The four determintants in second quantization langauge are \(\left|1100\right>\), \(\left|1001\right>\), \(\left|0110\right>\), and,...
In a previous post the Hellmann-Feynman forces where calculated using PySCF, and used gradient descent to optimize the geometry of the molecule. Geometry optimization is however a surprising hard problem to do, and writing algorithms from scratch might result in...
The Hellmann-Feynman theorem states that: \[\frac{\mathrm{d}E}{\mathrm{d\lambda}} = \left<\psi\left|\frac{\mathrm{d}\hat{H}}{\mathrm{d}\lambda}\right|\psi\right>\] It should be noted that for a finite basis this theorem does not hold, and might want to account for Pulay forces. For the molecular Hamiltonian: \[\hat{H} = -\frac{1}{2}\sum_i\nabla_i^2 - \sum_{iK}\frac{Z_K}{r_{iK}} +...
Current quantum computers work with unitary gates. In order to represent operators relevant for quantum chemistry on a quantum computer, unitary operators are thus needed. The condition for a unitary operator is: \[UU^\dagger = I\] It can easily be checked...
Coming from a theoretical chemistry background, I have mostly learned second quantization from Molecular Electronic-Structure Theory. From here the algebra is taught with the ON vector, \(\left|10\right>\) denoting the first orbital being occupied and the second orbital being unoccupied. In...
Following the source: A. Rizzo, S. Coriani, and K. Ruud, Computational Strategies for Spectroscopy. From Small Molecules to Nano Systems, edited by V. Barone (John Wiley and Sons, 2012) Chap. 2, pp.77–135. It is given in Eq. (2.35) that: \[\varepsilon(\omega)...
In quantum chemstry integrals of the following form are sometimes needed: \[F_a = \left<\phi_i\left|\nabla^a r^{-1}\right|\phi_j\right>\] But these integrals are often not directly available. What instead is available is integrals of the form: \[f_a = \left<\nabla^a\phi_i\left|r^{-1}\right|\phi_j\right>\] \[f_{a,b} = \left<\nabla^a\phi_i\left|r^{-1}\right|\nabla^b\phi_j\right>\] Let us...
Consider the electrostatic potential due to multipoles places at the position of atoms: \[E_{i}=\sum_{j}^{atom}\sum_{n}^{multipole}\frac{\left(-1\right)^{n}}{n!}T_{ij}^{(n)}m_{j}^{(n)}\] If the quantum mechanical electrostatic potential is to be minimized, a cost function can written of the form: \[z=\sum_{i}^{point}\left(V_{i,\mathrm{QM}}-E_{i}\right)^{2}+\sum_{l}^{constraints}\lambda_{l}g_{l}\] Expanding the square: \[z=\sum_{i}^{point}\left(V_{i,\mathrm{QM}}^{2}+E_{i}^{2}-2E_{i}V_{i,\mathrm{QM}}\right)+\sum_{l}^{constraints}\lambda_{l}g_{l}\] It is known...