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We say a system has multireference character if it is not well described at the SCF level by a single configuration/Slater determinant. Not accounting for this can lead to errors in post-SCF methods, which often rely on expansions of additional configurations generated from the SCF solution. It is fairly easy that this sort of error has occurred once a post-SCF solution has been constructed, but are there ways to predict this without performing a high level calculation?

I would be interested in both heuristic methods (e.g. compounds containing atom X or structure Y tend to have issues) as well as more quantitative approaches (e.g. X property from a DFT calculation is a good diagnostic for multireference character).

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This is not going to be a full answer, but maybe a starting point, a pointer to somewhat troubled systems. The general keyword here is degeneracy. Whenever you have fewer electrons than fit in the degenerate orbitals, two things may happen: In the simple case the molecule will distort, see aromaticity and anti-aromaticity. In the complicated case, you guessed it, you'll have a multireference system. A popular example for this is the rotational barrier of ethene, where you have degenerate, but orthogonal orbitals at the 90° dihedral. Two electrons, two orbitals, forced singlet will lead to no solution. If you treat it with TCSCF (two-configuration SCF) you'll get a qualitative solution, similar for CAS[2,2], and even UHF.

So the short and very unsatisfactory answer is, whenever you see a high symmetry, you should be careful.

Another obvious point is: anything with transition metals. Partially unoccupied d-orbitals will almost always need multi-reference treatment. In other words: Transition-metal chemistry is a graveyard for MP2 (large pdf via the Internet Archive)), dito in Koch/Holthausen's DFT book.

Some practical approaches:

  1. Calculate the fractional occupation number weighted electron density (FOD). This is fast and can be done at the semi-empirical level.[1] I don't think there is a faster (practical) way of doing this. (I cannot believe I forgot about this before.)

  2. A more conventional approach: UHF for an initial guess will give you approximate natural orbitals. You can use this to determine an active space, if you are so inclined.[2]

  3. UDFA and stability checks. Most Density Functional Methods are surprisingly robust about multi-reference character. If you run a calculation, then perform a stability analysis, which runs into a broken symmetry solution, you'll need to do multi-reference wave function methods.
    In general, performing a cheap method, will likely give you a HOMO-LUMO gap. When even your program of choice warns you about it being small, you might want to check for MR character.[3]

Obviously, your mileage may vary, and you need to check what is feasible for your system.

  1. Resources:
  2. See Chemistry.se for some more details: Is it reasonable to use natural orbitals of an unrestricted HF calculation as a basis to start a CASSCF calculation for a radical
  3. Some related questions on Chemistry.se (I know this gets old, but that's where the resources are.):
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As Martin has said a practical way to decide if a molecule requires multi-reference treatment is to calculate the unrestricted Hartree Fock natural orbitals.[1] Significant fractional occupancy of a UHF natural orbital indicates that the orbital should be included in the active space.

For a nonorthogonal finite basis set, they are given as

$$\mathbf{S}(\mathbf{D_\alpha+\mathbf{D_\beta}})\mathbf{S}\mathbf{C} = \mathbf{S}\mathbf{C}\mathbf{n},$$

where $\mathbf{S}$ is the overlap matrix, $\mathbf{D}_\alpha$ and $\mathbf{D}_\beta$ are the reduced first order density matrices for the $\alpha$ and $\beta$ spins, respectively, the columns of $\mathbf{C}$ contain the coefficients of the natural orbitals, and the diagonal matrix $\mathbf{n}$ holds the occupation numbers.

Specifically wavefunction with significant fractional occupancy (roughly between 0.02 and 1.98) should constitute the active space for static (non-dynamical) correlation.

For example, this can be performed in Molpro as 

***, HF
symmetry,nosym 
memory,400,m
orient,noorient

basis=6-31G**

geometry = {
N 0 0 0
O 0 0 1.75
}

{UHF;save,2101.2}
{matrop
load,D,Den
load,S
natorb,Cnat,D
mult,SC,S,Cnat
tran,D_nat,D,SC  ! = SC' D SC
prid,D_nat
save,Cnat,2150.2
}
{put,molden,orbitals.gmolden;orbital,2101.2}
{put,molden,check.gmolden;orbital,2150.2}

Molden will automatically print the occupations in the orbital dump (2101.2) but the matrop can be also used to solve the eigenvalue problem manually

The result for example here is 

   1.99999995   1.99999988   1.99987846   1.99937413   1.99796035   1.96525179   1.96022720   1.00000000   0.03977280   0.03474821   0.00203965   0.00062587   0.00012154   0.00000012   0.00000005   0.00000000  -0.00000000  -0.00000000  -0.00000000  -0.00000000  -0.00000000  -0.00000000  -0.00000000  -0.00000000  -0.00000000  -0.00000000   0.00000000   0.00000000

Which is in agreement with reference [1].

(If you do plan to do multi-reference however, do never use Hartree-Fock orbitals. They are the worst! Use DFT orbitals as a starting place.)

References:

  1. Pulay, P.; Hamilton, T. P. UHF natural orbitals for defining and starting MC‐SCF calculations. J. Chem. Phys. 1988, 88 (8), 4926–4933. DOI: 10.1063/1.454704.
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