# Molecular (infrared) vibrations: General relationship between molecular mechanics force constants and harmonic frequencies (in internal coordinates)?

In molecular mechanics method simple molecules, say symmetrical tri-atomics (H$_2$O), have potential energy surface governed by distances $r_1,r_2$ and angle $a$ like

$V (r_1,r_2,a_{121}) = K (r_1 - r_0)^2 + K (r_2 - r_0)^2 + K_a (a - a_0)^2$

The non-linear tri-atomic will have 3 harmonic frequencies $f_1,f_2,f_3$.

I understand that Hessian matrix seems to be important. The Hessian elements is $\frac{\partial^2 V}{\partial r_1^2}=2K$ and $\frac{\partial^2 V}{\partial r_2^2}=2K$ and $\frac{\partial^2 V}{\partial a_{121}^2}=2K_a$.

Is there a general relationship how harmonic frequencies are related to force constants? Does this relationship involve further approximations (besides the simple molecular mechanics PES)? The variables are internal coordinates and not the normal coordinates, right?

The water case I have just taken as illustrative example...

Thank you in advancement! Miranda Christina

• Several textbooks describe how to do this, so rather than repeat pages of this stuff have a look at the GF method (or FG method) on wikipedia, but more detailed examples are found in 'Atkins & Friedman 'Molecular Quantum Mechanics'; Graybeal, ' Molecular Spectroscopy'; Beddard 'Applying Maths in Chemical & Bimolecular Sciences'; Herzberg 'Infra Red & Raman Spectroscopy'. – porphyrin Aug 10 '16 at 17:57