4.4 SCF Initial Guess

4.4.1 Introduction

The Roothaan-Hall and Pople-Nesbet equations of SCF theory are non-linear in the molecular orbital coefficients. Like many mathematical problems involving non-linear equations, prior to the application of a technique to search for a numerical solution, an initial guess for the solution must be generated. If the guess is poor, the iterative procedure applied to determine the numerical solutions may converge very slowly, requiring a large number of iterations, or at worst, the procedure may diverge.

Thus, in an ab initio SCF procedure, the quality of the initial guess is of utmost importance for (at least) two main reasons:

To ensure that the SCF converges to an appropriate ground state. Often SCF calculations can converge to different local minima in wave function space, depending upon which part of “LCAO space” in which the initial guess lands.
When considering jobs with many basis functions requiring the recalculation of ERIs at each iteration, using a good initial guess that is close to the final solution can reduce the total job time significantly by decreasing the number of SCF iterations.

For these reasons, sooner or later most users will find it helpful to have some understanding of the different options available for customizing the initial guess. Q-Chem currently offers six options for the initial guess:

Superposition of Atomic Density (SAD)
Purified SAD guess (provides molecular orbitals; SADMO)
Core Hamiltonian (CORE)
Generalized Wolfsberg-Helmholtz (GWH)
Reading previously obtained MOs from disk. (READ)
Basis set projection (BASIS2)

The first four of these guesses are built-in, and are briefly described in Section 4.4.2. The option of reading MOs from disk is described in Section 4.4.3. The initial guess MOs can be modified, either by mixing, or altering the order of occupation. These options are discussed in Section 4.4.4. Finally, Q-Chem’s novel basis set projection method is discussed in Section 4.4.5.

4.4.2 Simple Initial Guesses

There are four simple initial guesses available in Q-Chem. While they are all simple, they are by no means equal in quality, as we discuss below.

Superposition of Atomic Densities (SAD): The SAD guess is almost trivially constructed by summing together atomic densities that have been spherically averaged to yield a trial density matrix. The SAD guess is far superior to the other two options below, particularly when large basis sets and/or large molecules are employed. There are three issues associated with the SAD guess to be aware of:
1. No molecular orbitals are obtained, which means that SCF algorithms requiring orbitals (the direct minimization methods discussed in Section 4.5) cannot directly use the SAD guess, and,
2. The SAD guess is not available for general (read-in) basis sets. All internal basis sets support the SAD guess.
3. The SAD guess is not idempotent and thus requires at least two SCF iterations to ensure proper SCF convergence (idempotency of the density).
Purified Superposition of Atomic Densities (SADMO): This guess is similar to the SAD guess, with two critical differences, namely, the removal of issues 1 and 3 above. The functional difference to the SAD guess is that the density matrix obtained from the superposition is diagonalized to obtain natural molecular orbitals, after which an idempotent density matrix is created by aufbau occupation of the natural orbitals. Since the initial density matrix is created with the SAD guess, the SADMO guess is not available either for a general (read-in) basis set.
Generalized Wolfsberg-Helmholtz (GWH): The GWH guess procedure[Wolfsberg and Helmholtz(1952)] uses a combination of the overlap matrix elements in Eq. eq411, and the diagonal elements of the Core Hamiltonian matrix in Eq. eq417. This initial guess is most satisfactory in small basis sets for small molecules. It is constructed according to the relation given below, where $c_ x$ is a constant typically chosen as $c_ x = 1.75$ .

$\begin{equation} \label{eq444} H_{\mu \upsilon } =c_ x S_{\mu \upsilon } (H_{\mu \mu } +H_{\upsilon \upsilon }) / 2. \end{equation}$ (4.32)
Core Hamiltonian: The core Hamiltonian guess simply obtains the guess MO coefficients by diagonalizing the core Hamiltonian matrix in Eq. (4.20). This approach works best with small basis sets, and degrades as both the molecule size and the basis set size are increased.

The selection of these choices (or whether to read in the orbitals) is controlled by the following $rem variables:

SCF_GUESS

Specifies the initial guess procedure to use for the SCF.

TYPE:

STRING

DEFAULT:

SAD

Superposition of atomic densities (available only with standard basis sets)

GWH

For ROHF where a set of orbitals are required.

FRAGMO

For a fragment MO calculation

OPTIONS:

CORE

Diagonalize core Hamiltonian

SAD

Superposition of atomic density

SADMO

Purified superposition of atomic densities (available only with standard basis sets)

GWH

Apply generalized Wolfsberg-Helmholtz approximation

READ

Read previous MOs from disk

FRAGMO

Superimposing converged fragment MOs

RECOMMENDATION:

SAD or SADMO guess for standard basis sets. For general basis sets, it is best to use the BASIS2 $rem. Alternatively, try the GWH or core Hamiltonian guess. For ROHF it can be useful to READ guesses from an SCF calculation on the corresponding cation or anion. Note that because the density is made spherical, this may favor an undesired state for atomic systems, especially transition metals. Use FRAGMO in a fragment MO calculation.

SCF_GUESS_ALWAYS

Switch to force the regeneration of a new initial guess for each series of SCF iterations (for use in geometry optimization).

TYPE:

LOGICAL

DEFAULT:

False

OPTIONS:

False

Do not generate a new guess for each series of SCF iterations in an

optimization; use MOs from the previous SCF calculation for the guess,

if available.

True

Generate a new guess for each series of SCF iterations in a geometry

optimization.

RECOMMENDATION:

Use the default unless SCF convergence issues arise

4.4.3 Reading MOs from Disk

There are two methods by which MO coefficients can be used from a previous job by reading them from disk:

Running two independent jobs sequentially invoking Q-Chem with three command line variables:.
```
   localhost-1> qchem job1.in job1.out save
   localhost-2> qchem job2.in job2.out save
```
Note: (1) The $rem variable SCF_GUESS must be set to READ in job2.in.
(2) Scratch files remain in $QCSCRATCH/save on exit.
Running a batch job where two jobs are placed into a single input file separated by the string @@@ on a single line.
Note: (1) SCF_GUESS must be set to READ in the second job of the batch file.
(2) A third qchem command line variable is not necessary.
(3) As for the SAD guess, Q-Chem requires at least two SCF cycles to ensure proper
SCF convergence (idempotency of the density).

Note: It is up to the user to make sure that the basis sets match between the two jobs. There is no internal checking for this, although the occupied orbitals are re-orthogonalized in the current basis after being read in. If you want to project from a smaller basis into a larger basis, consult section 4.4.5.

4.4.4 Modifying the Occupied Molecular Orbitals

It is sometimes useful for the occupied guess orbitals to be other than the lowest $N_{\alpha }$ (or $N_{\beta }$ ) orbitals. Reasons why one may need to do this include:

To converge to a state of different symmetry or orbital occupation.
To break spatial symmetry.
To break spin symmetry, as in unrestricted calculations on molecules with an even number of electrons.

There are two mechanisms for modifying a set of guess orbitals: either by SCF_GUESS_MIX, or by specifying the orbitals to occupy. Q-Chem users may define the occupied guess orbitals using the $occupied or $swap_occupied_virtual keywords. In the former, occupied guess orbitals are defined by listing the $\alpha$ orbitals to be occupied on the first line and $\beta$ on the second. In the former, only pair of orbitals that needs to be swapped is specified.

Note: (1) To prevent Q-Chem to change orbital occupation during SCF procedure, MOM_START option is often used in combination with $occupied or $swap_occupied_virtual keywords.
(2) The need for orbitals renders these options incompatible with the SAD guess. Most often, they are used with SCF_GUESS = READ.

Example 4.21 Format for modifying occupied guess orbitals.

$occupied
   1  2  3  4 ...  NAlpha
   1  2  3  4 ...  NBeta  
$end

Example 4.22 Alternative format for modifying occupied guess orbitals.

$swap_occupied_virtual
   <spin> <io1> <iv1>
   <spin> <io2> <iv2>
$end

Example 4.23 Example of swapping guess orbitals.

$swap_occupied_virtual
   alpha  5  6   
   beta   6  7   
$end

This is identical to:

Example 4.24 Example of specifying occupied guess orbitals.

$occupied
   1 2 3 4 6 5 7   
   1 2 3 4 5 7 6
$end

Example 4.25 Example of specifying occupied guess orbitals.

$occupied
   1:4 6 5 7   
   1:5 7 6
$end

The other $rem variables related to altering the orbital occupancies are:

SCF_GUESS_PRINT

Controls printing of guess MOs, Fock and density matrices.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0

Do not print guesses.

SAD

1

Atomic density matrices and molecular matrix.

2

Level 1 plus density matrices.

CORE and GWH

1

No extra output.

2

Level 1 plus Fock and density matrices and, MO coefficients and

eigenvalues.

READ

1

No extra output

2

Level 1 plus density matrices, MO coefficients and eigenvalues.

RECOMMENDATION:

None

SCF_GUESS_MIX

Controls mixing of LUMO and HOMO to break symmetry in the initial guess. For unrestricted jobs, the mixing is performed only for the alpha orbitals.

TYPE:

INTEGER

DEFAULT:

0 (FALSE)

Do not mix HOMO and LUMO in SCF guess.

OPTIONS:

0 (FALSE)

Do not mix HOMO and LUMO in SCF guess.

1 (TRUE)

Add 10% of LUMO to HOMO to break symmetry.

$n$

Add $n\times 10\%$ of LUMO to HOMO ( $0<n<10$ ).

RECOMMENDATION:

When performing unrestricted calculations on molecules with an even number of electrons, it is often necessary to break alpha/beta symmetry in the initial guess with this option, or by specifying input for $occupied.

4.4.5 Basis Set Projection

Q-Chem also includes a novel basis set projection method developed by Dr Jing Kong of Q-Chem Inc. It permits a calculation in a large basis set to bootstrap itself up via a calculation in a small basis set that is automatically spawned when the user requests this option. When basis set projection is requested (by providing a valid small basis for BASIS2), the program executes the following steps:

A simple DFT calculation is performed in the small basis, BASIS2, yielding a converged density matrix in this basis.
The large basis set SCF calculation (with different values of EXCHANGE and CORRELATION set by the input) begins by constructing the DFT Fock operator in the large basis but with the density matrix obtained from the small basis set.
By diagonalizing this matrix, an accurate initial guess for the density matrix in the large basis is obtained, and the target SCF calculation commences.

Two different methods of projection are available and can be set using the BASISPROJTYPE $rem. The OVPROJECTION option expands the MOs from the BASIS2 calculation in the larger basis, while the FOPPROJECTION option constructs the Fock matrix in the larger basis using the density matrix from the initial, smaller basis set calculation. Basis set projection is a very effective option for general basis sets, where the SAD guess is not available. In detail, this initial guess is controlled by the following $rem variables:

BASIS2

Sets the small basis set to use in basis set projection.

TYPE:

STRING

DEFAULT:

No second basis set default.

OPTIONS:

Symbol. Use standard basis sets as per Chapter 8.

BASIS2_GEN

General BASIS2

BASIS2_MIXED

Mixed BASIS2

RECOMMENDATION:

BASIS2 should be smaller than BASIS. There is little advantage to using a basis larger than a minimal basis when BASIS2 is used for initial guess purposes. Larger, standardized BASIS2 options are available for dual-basis calculations (see Section 4.7).

BASISPROJTYPE

Determines which method to use when projecting the density matrix of BASIS2

TYPE:

STRING

DEFAULT:

FOPPROJECTION (when DUAL_BASIS_ENERGY=false)

OVPROJECTION (when DUAL_BASIS_ENERGY=true)

OPTIONS:

FOPPROJECTION

Construct the Fock matrix in the second basis

OVPROJECTION

Projects MOs from BASIS2 to BASIS.

RECOMMENDATION:

None

Note: BASIS2 sometimes affects post-Hartree-Fock calculations. It is recommended to split such jobs into two subsequent one, such that in the first job a desired Hartree-Fock solution is found using BASIS2, and in the second job, which performs a post-HF calculation, SCF_GUESS=READ is invoked.

4.4.6 Examples

Example 4.26 Input where basis set projection is used to generate a good initial guess for a calculation employing a general basis set, for which the default initial guess is not available.

$molecule
   0  1
   O
   H  1  r
   H  1  r  2  a

   r    0.9
   a  104.0
$end

$rem
   METHOD        mp2
   BASIS         general
   BASIS2        sto-3g
$end

$basis
   O   0
   S   3   1.000000
           3.22037000E+02   5.92394000E-02
           4.84308000E+01   3.51500000E-01
           1.04206000E+01   7.07658000E-01
   SP  2   1.000000
           7.40294000E+00  -4.04453000E-01   2.44586000E-01
           1.57620000E+00   1.22156000E+00   8.53955000E-01
   SP  1   1.000000
           3.73684000E-01   1.00000000E+00   1.00000000E+00
   SP  1   1.000000
           8.45000000E-02   1.00000000E+00   1.00000000E+00
****
   H   0
   S   2   1.000000
           5.44717800E+00   1.56285000E-01
           8.24547000E-01   9.04691000E-01
   S   1   1.000000
           1.83192000E-01   1.00000000E+00
****
$end

Example 4.27 Input for an ROHF calculation on the OH radical. One SCF cycle is initially performed on the cation, to get reasonably good initial guess orbitals, which are then read in as the guess for the radical. This avoids the use of Q-Chem’s default GWH guess for ROHF, which is often poor.

$comment
   OH radical, part 1. Do 1 iteration of cation orbitals.
$end

$molecule
   1  1
   O  0.000   0.000   0.000
   H  0.000   0.000   1.000
$end

$rem
   BASIS           =  6-311++G(2df)
   METHOD          =  hf
   MAX_SCF_CYCLES  =  1
   THRESH          =  10
$end

@@@

$comment
   OH radical, part 2. Read cation orbitals, do the radical
$end

$molecule
   0  2
   O  0.000   0.000   0.000
   H  0.000   0.000   1.000
$end

$rem
   BASIS            =  6-311++G(2df)
   METHOD           =  hf
   UNRESTRICTED     =  false
   SCF_ALGORITHM    =  dm
   SCF_CONVERGENCE  =  7
   SCF_GUESS        =  read
   THRESH           =  10
$end

Example 4.28 Input for an unrestricted HF calculation on H $_{2}$ in the dissociation limit, showing the use of SCF_GUESS_MIX = 2 (corresponding to 20% of the alpha LUMO mixed with the alpha HOMO). Geometric direct minimization with DIIS is used to converge the SCF, together with MAX_DIIS_CYCLES = 1 (using the default value for MAX_DIIS_CYCLES, the DIIS procedure just oscillates).

$molecule
   0  1
   H  0.000   0.000   0.0
   H  0.000   0.000 -10.0
$end

$rem
   UNRESTRICTED     =  true
   METHOD           =  hf
   BASIS            =  6-31g**
   SCF_ALGORITHM    =  diis_gdm
   MAX_DIIS_CYCLES  =  1
   SCF_GUESS        =  gwh
   SCF_GUESS_MIX    =  2
$end

SAD	Superposition of atomic densities (available only with standard basis sets)
GWH	For ROHF where a set of orbitals are required.
FRAGMO	For a fragment MO calculation

CORE	Diagonalize core Hamiltonian
SAD	Superposition of atomic density
SADMO	Purified superposition of atomic densities (available only with standard basis sets)
GWH	Apply generalized Wolfsberg-Helmholtz approximation
READ	Read previous MOs from disk
FRAGMO	Superimposing converged fragment MOs

False	Do not generate a new guess for each series of SCF iterations in an
	optimization; use MOs from the previous SCF calculation for the guess,
	if available.
True	Generate a new guess for each series of SCF iterations in a geometry
	optimization.

0	Do not print guesses.
SAD
1	Atomic density matrices and molecular matrix.
2	Level 1 plus density matrices.
CORE and GWH
1	No extra output.
2	Level 1 plus Fock and density matrices and, MO coefficients and
	eigenvalues.
READ
1	No extra output
2	Level 1 plus density matrices, MO coefficients and eigenvalues.

0 (FALSE)	Do not mix HOMO and LUMO in SCF guess.
1 (TRUE)	Add 10% of LUMO to HOMO to break symmetry.
$n$	Add $n\times 10\%$ of LUMO to HOMO ( $0<n<10$ ).

Symbol. Use standard basis sets as per Chapter 8.
BASIS2_GEN	General BASIS2
BASIS2_MIXED	Mixed BASIS2

FOPPROJECTION	Construct the Fock matrix in the second basis
OVPROJECTION	Projects MOs from BASIS2 to BASIS.