How the molecules optimization process done by quantum chemistry software ?

Joao.T

Hi, I'm João. I'm first year undergraduate student inDepartment of Fundamental Chemistry at University of São Paulo.

I have a keen interest in theoretical and computational chemistry, that's why I started to learn by myself in this field with individual effort .
Recently, i faced a difficulty to understand the optimization process of a given molecule and find a stationary point, as i know to find the minimum, the derivative of energy must be equal zero,
which means the energy is a function, what i want to understand is the mathematical expression of this function, and how quantum chemistry softwares optimize a molecule by by given only the cartesian coordinates of this molecule ? I know there there is algorithm for optimization such as Newton Raphson, Conjugate Gradient and steepest descent, but, how the derivatives have been done for a molecule has only the cartesian coordinates ?
Just to mention, I tried to find the function energy of a simple molecule like a water molecule and discovered the process of calculation manually but I failed. So many question inside my head without finding any answer, i hope i could find what i'm looking for in this wonderful forum.


So many question inside my head without finding any answer ,i hope i could find what i'm looking for in this wonderful forum.

wzkchem5

Briefly speaking, the energy function has no closed-form expression (unless you are using a molecular mechanics method, in which case some typical function forms can be found in https://en.wikipedia.org/wiki/Force_field_(chemistry) ).
In ab initio methods, the energy E({R}, Psi) is an explicit function of the atomic coordinates {R} and the wavefunction Psi. If for every possible {R}, you tune Psi so that E({R}, Psi) is minimized (but subject to some constraints, like Psi is normalized and has the correct number of alpha and beta electrons; with some methods you have further constraints like Psi is representable by a single Slater determinant, etc.), then you get a function U({R}) that only depends on {R} (because for each {R} you manually specified a Psi), and this is the function that you are optimizing in a geometry optimization. However it is an implicit function, because the Psi that minimizes E({R}, Psi) at a given {R} is an implicit function of {R}. There are also some methods (so-called non-variational methods) where Psi is not found by minimizing E({R}, Psi), but for the moment you can temporarily ignore them. An introductory read on this can be found at https://en.wikipedia.org/wiki/Bo ... imer_approximation.

The derivatives of U({R}) with respect to the atomic coordinates {R} are usually found by the chain rule, from partial derivatives of E({R}, Psi) with respect to {R} and Psi. For variational methods (like HF, DFT and CASSCF) the partial derivatives of E({R}, Psi) with respect to Psi are zero, since E({R}, Psi) has been minimized with respect to Psi, and this simplifies the formalism. For this part you may have a look at the DFT energy gradient article https://aip.scitation.org/doi/10.1063/1.464906, although you may want to start from the derivation of the HF gradient, which is in an earlier article cited therein.

北大-陶豫

As to force (1st order deriv). There's a Hellmann–Feynman theorem that enables us to obtain the exact gradient from the exact wavefunction. In practise there are some details however. For example, some variations are used for HF, DFT, etc since they don't have the exact wavefunction. Also there is a Pulay force due to incompleteness of basis set. In quantum chemistry programs, usually most methods (HF, DFT, MP2, CASSCF, CISD, CCSD, etc) have analytical gradients (i. e. no need to compute numerical gradient, which does finite difference on the coordinates of nuclears, slower and less accurate).

Analytical 2nd order deriv is also available for many methods. For HF and DFT it's obtained by solving CPHF equation. It's much more difficult to obtain the analytical 2nd order deriv than the analytical 1st order deriv.

In short: Although the exact analytical form of a PES at a given level is unavailable, its 1st order deriv is usually available through Hellmann–Feynman theorem and its variations, and higher order deriv are also available sometimes.

Attached is a homework I did in my undergraduate years. It's about the PES of water molecule, and I optimized the molecule based on the PES. Hundreds of parameters were used to fit the PES (note that such an analytical expression is unavailable in most most cases).

sobereva

I strongly suggest that you first understand how the force of Hartree-Fock is calculated, this is much simpler compared to other situations (DFT, CI, CC, etc.). In page 440 of "Modern Quantum Chemistry" by Szabo there is very clear derivation of the formulae used to evaluate analytic gradient of Hartree-Fock, this a nice starting point of the topic you are interested in.

For force evaluation of LDA and GGA types of DFT functionals, see Chem. Phys. Lett., 199, 557 (1992).

For more information about derivative evaluations, see Section 11.3 of Introduction to Computational Chemistry (3ed,Frank Jensen,2017).