4.9 Unconventional SCF Calculations 4.9.1 Polarized Atomic Orbital (PAO) Calculations 4.9.3 Multiple SCF Solutions for Non-Orthogonal CI

4.9.2 SCF Metadynamics

(September 1, 2024)

As the SCF equations are non-linear in the electron density, there are in theory very many solutions, i.e., sets of orbitals where the energy is stationary with respect to changes in the orbital subset. Most often sought is the solution with globally minimal energy as this is a variational upper bound to the true eigenfunction in this basis. The SCF methods available in Q-Chem allow the user to converge upon an SCF solution, and (using STABILITY_ANALYSIS) ensure it is a minimum, but there is no known method of ensuring that the found solution is a global minimum; indeed in systems with many low-lying energy levels the solution converged upon may vary considerably with initial guess.

SCF metadynamics ¹²⁶⁴ Thom A. J. W., Head-Gordon M.
Phys. Rev. Lett.
(2008), 101, pp. 193001. Link is a technique which can be used to locate multiple SCF solutions, and thus gain some confidence that the calculation has converged upon the global minimum. It works by searching out a solution to the SCF equations. Once found, the solution is stored, and a biasing potential added so as to avoid re-converging to the same solution. More formally, the distance between two solutions, $w$ and $x$ , can be expressed as $d_{wx}^{2}=\langle{{}^{w}\!\Psi|^{w}\!\hat{\rho}-{{}^{x}}\!\hat{\rho}}|^{w}\!\Psi\rangle$ , where ${}^{w}\!\Psi$ is a Slater determinant formed from the orthonormal orbitals, ${}^{w}\!\phi_{i}$ , of solution $w$ , and ${}^{w}\!\hat{\rho}$ is the one-particle density operator for ${}^{w}\!\Psi$ . This definition is equivalent to $d_{wx}^{2}=N-{{}^{w}}\!P^{\mu\nu}S_{\nu\sigma}\cdot{{}^{x}}\!P^{\sigma\tau}S_{% \tau\mu}.$ and is easily calculated. The function $d_{wx}^{2}$ is between zero and the number of electrons, and can be taken as the distance between two solutions. As an example, any singly-excited determinant (which will not in general be another SCF solution) is a distance 1 away from the reference (unexcited) determinant.

In a manner analogous to classical metadynamics, to bias against the set of previously located solutions, $x$ , we create a new Lagrangian,

\displaystyle\tilde{E}=E+\sum_{x}N_{x}e^{-\lambda_{x}d_{0x}^{2}}

(4.63)

where $0$ represents the present density. From this we may derive a new effective Fock matrix,

\displaystyle\tilde{F}_{\mu\nu}=F_{\mu\nu}+\sum_{x}{{}^{x}\!P}_{\mu\nu}N_{x}% \lambda_{x}e^{-\lambda_{x}d_{0x}^{2}}

(4.64)

This may be used with very little modification within a standard DIIS procedure to locate multiple solutions. When close to a new solution, the biasing potential is removed so the location of that solution is not affected by it. If the calculation ends up re-converging to the same solution, $N_{x}$ and $\lambda_{x}$ can be modified to avert this. Once a solution is found it is added to the list of solutions, and the orbitals mixed to provide a new guess for locating a different solution.

This process can be customized by the REM variables below. Both DIIS and GDM methods can be used, but it is advisable to turn on MOM when using DIIS to maintain the orbital ordering. Post-HF correlation methods can also be applied. By default they will operate for the last solution located, but this can be changed with the SCF_MINFIND_RUNCORR variable.

SCF_SAVEMINIMA

SCF_SAVEMINIMA
       Turn on SCF metadynamics and specify how many solutions to locate.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Do not use SCF metadynamics $n$ Attempt to find $n$ distinct SCF solutions.
RECOMMENDATION:
       Perform SCF Orbital metadynamics and attempt to locate $n$ different SCF solutions. Note that these may not all be minima. Many saddle points are often located. The last one located will be the one used in any post-SCF treatments. In systems where there are infinite point groups, this procedure cannot currently distinguish between spatial rotations of different densities, so will likely converge on these multiply.

SCF_READMINIMA

SCF_READMINIMA
       Read in solutions from a previous SCF metadynamics calculation
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       $n$ Read in $n$ previous solutions and attempt to locate them all. $-n$ Read in $n$ previous solutions, but only attempt to locate solution $n$ (not available in libnoci).
RECOMMENDATION:
       This may not actually locate all solutions required and will probably locate others too. The SCF will also stop when the number of solutions specified in SCF_SAVEMINIMA are found. Solutions from other geometries may also be read in and used as starting orbitals. If a solution is found and matches one that is read in within SCF_MINFIND_READDISTTHRESH, its orbitals are saved in that position for any future calculations. The algorithm works by restarting from the orbitals and density of a the minimum it is attempting to find. After 10 failed restarts (defined by SCF_MINFIND_RESTARTSTEPS), it moves to another previous minimum and attempts to locate that instead. If there are no minima to find, the restart does random mixing (with 10 times the normal random mixing parameter). Note that in libnoci, previous minima are read using NOCI_REFGEN = 1, whilst the exact solutions are specified as described in Section 4.9.3

SCF_MINFIND_WELLTHRESH

SCF_MINFIND_WELLTHRESH
       Specify what SCF_MINFIND believes is the basin of a solution
TYPE:
       INTEGER
DEFAULT:
       5
OPTIONS:
       $n$ for a threshold of $10^{-n}$
RECOMMENDATION:
       When the DIIS error is less than $10^{-n}$ , penalties are switched off to see whether it has converged to a new solution.

SCF_MINFIND_RESTARTSTEPS

SCF_MINFIND_RESTARTSTEPS
       Restart with new orbitals if no minima have been found within this many steps
TYPE:
       INTEGER
DEFAULT:
       300
OPTIONS:
       $n$ Restart after $n$ steps.
RECOMMENDATION:
       If the SCF calculation spends many steps not finding a solution, lowering this number may speed up solution-finding. If the system converges to solutions very slowly, then this number may need to be raised.

SCF_MINFIND_INCREASEFACTOR

SCF_MINFIND_INCREASEFACTOR
       Controls how the height of the penalty function changes when repeatedly trapped at the same solution
TYPE:
       INTEGER
DEFAULT:
       10100 meaning 1.01
OPTIONS:
       $a b c d e$ corresponding to $a . b c d e$
RECOMMENDATION:
       If the algorithm converges to a solution which corresponds to a previously located solution, increase both the normalization N and the width lambda of the penalty function there. Then do a restart.

SCF_MINFIND_INITLAMBDA

SCF_MINFIND_INITLAMBDA
       Control the initial width of the penalty function.
TYPE:
       INTEGER
DEFAULT:
       02000 meaning 2.000
OPTIONS:
       $a b c d e$ corresponding to $a b . c d e$
RECOMMENDATION:
       The initial inverse-width (i.e., the inverse-variance) of the Gaussian to place to fill solution’s well. Measured in electrons ${}^{(}-1)$ . Increasing this will repeatedly converging on the same solution.

SCF_MINFIND_INITNORM

SCF_MINFIND_INITNORM
       Control the initial height of the penalty function.
TYPE:
       INTEGER
DEFAULT:
       01000 meaning 1.000
OPTIONS:
       $a b c d e$ corresponding to $a b . c d e$
RECOMMENDATION:
       The initial normalization of the Gaussian to place to fill a well. Measured in hartrees.

SCF_MINFIND_RANDOMMIXING

SCF_MINFIND_RANDOMMIXING
       Control how to choose new orbitals after locating a solution
TYPE:
       INTEGER
DEFAULT:
       00200 meaning .02 radians
OPTIONS:
       $a b c d e$ corresponding to $a . b c d e$ radians
RECOMMENDATION:
       After locating an SCF solution, the orbitals are mixed randomly to move to a new position in orbital space. For each occupied and virtual orbital pair picked at random and rotate between them by a random angle between 0 and this. If this is negative then use exactly this number, e.g., $-15708$ will almost exactly swap orbitals. Any number $<-15708$ will cause the orbitals to be swapped exactly.

SCF_MINFIND_NRANDOMMIXES

SCF_MINFIND_NRANDOMMIXES
       Control how many random mixes to do to generate new orbitals
TYPE:
       INTEGER
DEFAULT:
       10
OPTIONS:
       $n$ Perform $n$ random mixes.
RECOMMENDATION:
       This is the number of occupied/virtual pairs to attempt to mix, per separate density (i.e., for unrestricted calculations both alpha and beta space will get this many rotations). If this is negative then only mix the highest 25% occupied and lowest 25% virtuals.

SCF_MINFIND_READDISTTHRESH

SCF_MINFIND_READDISTTHRESH
       The distance threshold at which to consider two solutions the same
TYPE:
       INTEGER
DEFAULT:
       00100 meaning 0.1
OPTIONS:
       $a b c d e$ corresponding to $a b . c d e$
RECOMMENDATION:
       The threshold to regard a minimum as the same as a read in minimum. Measured in electrons. If two minima are closer together than this, reduce the threshold to distinguish them.

SCF_MINFIND_MIXMETHOD

SCF_MINFIND_MIXMETHOD
       Specify how to select orbitals for random mixing
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Random mixing: select from any orbital to any orbital. 1 Active mixing: select based on energy, decaying with distance from the Fermi level. 2 Active Alpha space mixing: select based on energy, decaying with distance from the Fermi level only in the alpha space.
RECOMMENDATION:
       Random mixing will often find very high energy solutions. If lower energy solutions are desired, use 1 or 2.

SCF_MINFIND_MIXENERGY

SCF_MINFIND_MIXENERGY
       Specify the active energy range when doing Active mixing
TYPE:
       INTEGER
DEFAULT:
       00200 meaning 00.200
OPTIONS:
       $a b c d e$ corresponding to $a b . c d e$
RECOMMENDATION:
       The standard deviation of the Gaussian distribution used to select the orbitals for mixing (centered on the Fermi level). Measured in Hartree. To find less-excited solutions, decrease this value

SCF_MINFIND_RUNCORR

SCF_MINFIND_RUNCORR
       Run post-SCF correlated methods on multiple SCF solutions
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       If this is set $>0$ , then run correlation methods for all found SCF solutions.
RECOMMENDATION:
       Post-HF correlation methods should function correctly with excited SCF solutions, but their convergence is often much more difficult owing to intruder states.

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4.9.2 SCF Metadynamics