# How is the Hessian evaluated?

I was reading about the various ways to optimise the geometry of the system, and found that none of the sources mention how the Hessian matrix is obtained.

How does one obtain the Hessian matrix? I’m thinking it’s either some analytical formula or just numerical differentiation.

• Both analytically and via numerical derivatives is possible and has been used extensively. However, for straight geometry optimization, one typically uses approximate forms of the Hessian (a pseudo-Hessian, if you will, built over several geometry optimization cycles). For transition state search, one usually requires the true Hessian. – TAR86 Sep 9 at 18:02
• What method are you talking about (ab initio electronic structure, atomistic MD)? What software? – Buck Thorn Sep 10 at 6:52
• I'm referring to ab initio electronic structure. – ANZGC FlyingFalcon Sep 10 at 13:43
• Are you looking for a detailed mathematical description or a more qualitative description of how the Hessian is estimated? – Buck Thorn Sep 11 at 7:32

The documentation for the Gaussian software is actually pretty complete, even if qualitative. On the subject of optimizations, I recommend you look up the freq and opt keywords.

The manual explains the procedure followed by the Berny optimization as involving construction of an initial analytical estimate of the Hessian using a simplified force field, with subsequent optimization steps improving on that estimate using computed gradients. A number of options are provided for improving the initial estimate of the force constants where the defaults derived from the estimated Hessian are not good enough.

For frequency calculations, accuracy requires a good estimate of the full Hessian, better than estimates that might suffice for geometry optimizations. Gaussian allows for analytical computation of force constants for some of the methods (e.g. RHF) and uses analytical gradients to compute a numerical Hessian with some other methods, or if requested. In optimization cases where convergence is an issue, Gaussian facilitates convergence to a real minimum by employing the Opt=CalcAll keyword, which results in analytical (where possible) computation of the full Hessian (at some cost though).

• Thanks! I'll certainly look it up. Could you also point me towards a mathematical explanation, if possible? – ANZGC FlyingFalcon Sep 12 at 12:59
• @ANZGCFlyingFalcon I can add a little info if it helps, when I get a chance. A numerical methods textbook that covers optimization methods should discuss this btw. – Buck Thorn Sep 12 at 13:01