Daily Archives: June 23, 2010
It’s always hard to write about a computational method in a blog post since it could span for pages on end, so that’s why this post as the ones about NBO and PCM will only treat briefly the underlying physics (at a handwave level) and interpretation about the most popular choices within multiconfigurational methods which form part of the family of post Hartree-Fock methods*. If you are only interested in one of the many MC methods scroll down the post to find the one you need. As usual this post may become updated/expanded in time so come back to see what is new, or better yet subscribe to this and other posts to get those updates straight to your inbox.
The first and most intuitive approach to electronic structure calculations still is the Slater determinant (SD) in which a single electronic configuration is taken into account while obeying the aufbau principle (filling of the orbitals from the core up) and Pauli’s exclusion principle (antisymmetry of the wavefunction respect to the exchange of two particles). This procedure, under the regular HF Hamiltonian, neglects the instantaneous interactions between electrons which accounts to some extent for electron correlation. In order to fix this shortcoming one might use a wavefunction that is a sum of two or more Slater determinants in which each one represents a different electron configuration. This second SD represents the promotion of an electron to a virtual orbital (normally the lowest lying one) and we may add another SD in which this electron is promoted to another virtual orbital and even more we can assume double excitations, that is the promotion of two electrons at once. If we were to take into consideration ALL possible combinations, we say we are dealing with the Complete Active Space (the CAS in CASSCF), of course the amount of computing time scales rapidly as we increase the active space so real CASSCF are practically unachievable. The inclusion of these subsequent SD’s corrects the wavefunction as to yield a better description of the electron system by taking into consideration all possible configurations. MC methods are needed not only to account for electron correlation but also to allow orbital readjustments to means of allowing polarized orbital pair formation.
It should be noticed that some of the following procedures while are variational (i.e. yield an upper bound to the total molecular energy) they are not size-extensive and therefore are unreliable in treating complex-associated systems or extended solids. Perturbation methods such as MPn and Coupled Cluster are not variational in nature but yield size-extensive energies.
Multiconfigurational self-consistent field (MCSCF)
When dealing with many determinants to optimize the function E, we run into the problem that convergence will be slow and sometimes located on local rather than on global minima. MCSCF wavefunctions are strongly dependant of the orbitals (or rather their coefficients) with high occupancy, while are least affected by those with lower occupancies. It is for this reason that the number of configurations in MCSCF is usually kept to a small to moderate number chosen to describe essential correlations and important dynamical corrlations. As a result the ad hoc wavefunction is designed to provide a good description of those orbitals with higher occupation numbers and neglects those that have low occupation numbers.
Configurations Interaction, CI
A set of orthonormal MO’s is first obtained from an SCF or a small MCSCF calculation. The orbitals thus obtained will be refered to as reference space. The coefficients of the LCAO expansion are no longer optimized but rather the Ci coefficients of the CI expansion, that is the coefficients with which each electron configuration is taken into account into the CI wavefunction, and for this reason one may include many more configurations than in the previous method. Of course the space can also be chosen in such a way that some excitations are neglected, for instance those arising from core electrons (a bond breaking process will be more affected by valence electrons than from those in the core).
Coupled Cluster, CC, CCS, CCSD, CCSDT
The CC methods rely on a far more elegant approach: the use of creation and anhiliation operators which eliminate an electron from a certain orbital while making it appear on another. For an extraordinary introduction to the underlying mathematics of these methods refer to the paper by H.F. Schaefer listed below under references. Because of the nature of the cluster operator T the CC equations are quartic and although it is a huge task to evaluate all of the commutator matrix elements, it can be done with rather available computational resources. In order to avoid convergence problems, non-linear terms should be neglected if possible. This a size extensive method but not a variational one.
Complete Active Space Self Consistent Field, CASSCF
CASSCF includes all configurations possible that can be created by distributing N valence electrons in M orbitals in such a manner that the spin is maintained constant. Clearly this makes the number of configurations scale rather rapidly so practical considerations dictate that CAS based approaches be limited to situations in which only a few electrons are to be correlated (hence using only a few orbitals).
Some common problems and their solutions
Which one to use? Of course it depends on the properties to be calculated as well as on the kind of system at hand. Some systems require size extensive methods (solids, aggregates, complexes, etc.) Other scenarios such as chemical reactions or bond breaking events require
Selection of the active space is critical!
All these methods are very demanding both in terms of computing time and memory available, which makes it very common to reach disk limits. In GaussianX you may see an error of the following sort:
“Erroneous write. write -1 instead of -1472711800” (numbers here are only examples)
I suggest to make use of %rwf capabilities whenever possible. Some clusters have queues with different memory sizes, try queuing your job to a queue with a larger disk. Maybe Gaussian is not the best way to go when it comes to multireference methods; if available try using MOLCAS or MOLPRO.
As usual feel free to comment and rate this post (criticism is also very much welcome, so please!) that allows me to know somebody is reading and further encourages me to keep this blog not only as a place I rant in soliloquies 😀
– Reviews in Computational Chemistry Vol.14 (Mathematical grounds for MC methods for beginners, if you are rusty on your matrix algebra or series, try brushing them up first) (Click here for an old post on Henry Schaefer in this same blog; don’t get me wrong! I like the guy’s science and I think he is brilliant but this is a controversial man and I wrote about that too, back in the days when posts on this blog were mostly short random thoughts).
-Gaussian’s tech support site.
-Quantum Mechanics in Chemistry, Jack Simons & Jeff Nichols (1997) Oxford University Press
*The name post HF method does not imply that the procedures are not applicable to DFT constructed wavefunctions (but I thought DFT fans didn’t like to work with wavefunctions? Well, so did I!)