The ALMO-EDA method introduced in Section 12.5 is a very useful tool for unraveling the nature of intermolecular interactions. Nevertheless, it has two major shortcomings: (i) Although the polarization (POL) energy is variationally evaluated, it does not have a meaningful basis set limit. As the employed basis set becomes larger, the POL term starts to be contaminated by charge transfer (CT) and loses its intended meaning. (ii) The frozen (FRZ) interaction is a monolithic term in the original ALMO-EDA scheme. In practice, further decomposition of the FRZ term is often desired. For example, if one wants to use ALMO-EDA as a tool for the development of empirical force fields, the separation of the FRZ term into contributions from permanent electrostatics, Pauli repulsion and dispersion will be helpful since they are usually modeled by distinct functional forms in classical force fields. These drawbacks have been addressed recently, defining the second generation of the ALMO-EDA method (also referred to as “EDA2" in the text below) [839, 841, 842].

The original definition of the ALMOs used in SCFMI calculations is based on the fragment-blocking structure of the AO-to-MO transformation matrix, *i.e.*, for a given fragment, the associated MOs can only be expanded by the AO basis functions centered on the atoms that belong to the same fragment. Here we propose a generalized definition for SCFMI calculations: given a set of basis vectors () in which each of them is tagged to a fragment but is allowed to be spanned by any AO basis function, it defines the working basis of the SCFMI problem. Then, within this basis, the locally projected SCF equations can be solved in a similar way, with the constraint that the MO coefficient matrix in the working basis () is fragment-block-diagonal, while the MO coefficient matrix in the AO basis does not necessarily retain the blocking structure. The basis vectors in can be either non-orthogonal or orthogonal between fragments. More details on the generalized SCFMI equations are available in ref. Horn:2015.

This generalized SCFMI scheme is implemented in GEN_SCFMAN (the original AO-block based scheme is available in GEN_SCFMAN as well). It is utilized for the variational optimization of the polarized (but CT-forbidden) intermediate state in “EDA2" (see Section 12.7.2). Another preferable feature of this generalized scheme is that the interfragment linear dependency in can be properly handled. Therefore, this scheme can be used to replace the original AO-block based SCFMI without becoming ill-defined when interfragment linear dependency occurs. In contrast, the original ALMO-EDA method that employs the AO-block based approach fails when the sum of the number of orbitals on each fragment is not equal to the number of orbitals for the super-system (the latter is determined by the total number of AO basis functions and BASIS_LIN_DEP_THRESH), which often happens when substantially large basis sets are used or when the super-system comprises a large number of fragments.

SCFMI calculations based on the GEN_SCFMAN implementation are triggered by setting GEN_SCFMAN = TRUE and FRGM_METHOD = STOLL or GIA (the other options of FRGM_METHOD are not allowed). A subset of supported algorithms in GEN_SCFMAN are available for restricted (R) and unrestricted (U) SCFMI, including DIIS, GDM_LS, and NEWTON_CG. While the DIIS algorithm iteratively solves for the locally-projected SCF equations, the latter two methods use the energy derivatives with respect to the on-fragment orbital rotations to minimize the energy until the gradient reaches zero. As for standard calculations using GEN_SCFMAN, internal stability analysis is also available for R- and U-SCFMI, and one can set FD_MAT_VEC_PROD to TRUE if the analytical Hessian is not available for the employed density functional.

As in the original implementation, perturbative corrections can be applied on top of the SCFMI solution to approach the full SCF result, and this is still controlled by FRGM_LPCORR. Note that among the options introduced in Section 12.6, only ARS and RS are allowed here since the exact SCF calculation is actually beyond the scope of SCFMI.

In addition, with this more general implementation users are allowed to specify some fragments to be frozen during the SCFMI calculation, *i.e.*, intrafragment relaxation does not occur on these fragments. This is achieved by specifying the *$rem* variable SCFMI_FREEZE_SS. Such a calculation can be interpreted as an active fragment being embedded in a frozen environment where the interaction between them is treated quantum mechanically.

SCFMI_MODE

Determine whether generalized SCFMI is used and also the property of the working basis.

TYPE:

INTEGER

DEFAULT:

0

AO-block based SCFMI (the original definition of ALMOs).

1

Generalized SCFMI with basis vectors that are non-orthogonal between fragments.

2

Generalized SCFMI with basis vectors that are orthogonal between fragments.

OPTIONS:

0 (“1" is used by basic “EDA2" calculations).

RECOMMENDATION:

None

SCFMI_FREEZE_SS

Keep the first several fragments unrelaxed in an SCFMI calculation.

TYPE:

INTEGER

DEFAULT:

Freeze the first fragments.

OPTIONS:

0 (all fragments are active)

RECOMMENDATION:

None

**Example 12.309** Generalized SCFMI calculation for the water dimer with single Roothaan-step perturbative correction. For this specific case, the result is identical to that given by AO-block based SCFMI (SCFMI_MODE = 0).

$molecule 0 1 -- 0 1 O -1.551007 -0.114520 0.000000 H -1.934259 0.762503 0.000000 H -0.599677 0.040712 0.000000 -- 0 1 O 1.350625 0.111469 0.000000 H 1.680398 -0.373741 -0.758561 H 1.680398 -0.373741 0.758561 $end $rem JOBTYPE sp METHOD b3lyp GEN_SCFMAN true BASIS 6-31+G(d) GEN_SCFMAN true SCF_CONVERGENCE 8 THRESH 14 SYMMETRY false SYM_IGNORE true FRGM_METHOD stoll FRGM_LPCORR rs SCFMI_MODE 1 !gen scfmi (non-orthogonal) $end

The definition of polarization energy lowering in the original ALMO-EDA utilizes the full AO space of each fragment as the variational degrees of freedom. This is based on the assumption that the AO basis functions are fragment-ascribable based on their atomic centers. However, this assumption becomes inappropriate when very large basis sets are used, especially those with diffuse functions (*e.g.* def2-QZVPPD). In such scenarios, basis functions on a given fragment tend to describe other fragments so that the “absolute localization" constraint becomes weaker and finally gets effectively removed. This is why the original ALMO-EDA scheme does not have a well-defined basis set limit for its polarization energy.

To overcome this problem, Horn and Head-Gordon proposed a new definition for the POL term in the ALMO-EDA method based on fragment electrical response functions (FERFs) [839]. FERFs on a given fragment are prepared by solving CPSCF equations after its SCF solution is found:

(12.1) |

where is SCF orbital Hessian and is a component () of a multipole matrix with a certain order. The resulting fragment response matrices () are a set of matrices. Then, a singular value decomposition (SVD) is performed on :

(12.2) |

and the left vectors (not including the null vectors) will be used to construct a truncated virtual space, which is used to define the variational degrees of freedom for the SCFMI problem:

(12.3) |

where denotes the original virtual orbitals of the given fragment.

The basic spirit of using FERFs is to obtain a subset of virtuals that is most pertinent to the electrical polarization of a given fragment, while the redundant variational degrees of freedom (which might be CT-like) are excluded. This scheme is shown to give a well-defined basis set limit for the polarization energy that relies on the SCFMI calculation. The multipole orders (dipole (D), quadrupole (Q), and octopole (O)) included on the RHS of eq. decide the span of FERFs on each fragment. Numerical experiments suggest that the inclusion of dipole- and quadrupole-type responses is able to long-range induced electrostatics correctly and also gives a well-defined basis set limit, which is thus recommended as the working basis of the SCFMI problem. The full span of the polarization subspace of fragment is thus:

(12.4) |

Therefore, each occupied orbital will be paired with eight virtual orbitals (if the employed AO basis is large enough).

The polarization subspaces constructed as in eq. are non-orthogonal between fragments. Therefore, it is named as the “nDQ" model for polarization. There is another version of this method which enforces interfragment orthogonality between the polarization subspaces and it is correspondingly termed as “oDQ" (or with other multipole orders). The preparation of orthogonal FERFs is more complicated (see ref. Horn:2015 for the details) and usually gives less favorable polarization energies. For most general cases, we recommend the use of the “nDQ" model. Calculations using FERFs are performed using the generalized SCFMI procedure introduced in Section 12.7.1.

CHILD_MP

Compute FERFs for fragments and use them as the basis for SCFMI calculations.

TYPE:

BOOLEAN

DEFAULT:

FALSE

OPTIONS:

FALSE

Do not compute FERFs (use the full AO span of each fragment).

TRUE

Compute fragment FERFs.

RECOMMENDATION:

Use FERFs to compute polarization energy when large basis sets are used. In an “EDA2" calculation, this

$remvariable is set based on the given option automatically.

CHILD_MP_ORDERS

The multipole orders included in the prepared FERFs. The last digit specifies how many multipoles to compute, and the digits in the front specify the multipole orders: 1: dipole (D); 2: quadrupole (Q); 3: octopole (O).

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

11

D

122

DQ

1233

DQO

RECOMMENDATION:

Use 122 (DQ) when FERF is needed.

**Example 12.310** Generalized SCFMI calculation for the water dimer using nDQ FERFs.

$molecule 0 1 -- 0 1 O -1.551007 -0.114520 0.000000 H -1.934259 0.762503 0.000000 H -0.599677 0.040712 0.000000 -- 0 1 O 1.350625 0.111469 0.000000 H 1.680398 -0.373741 -0.758561 H 1.680398 -0.373741 0.758561 $end $rem JOBTYPE sp METHOD wb97x-v GEN_SCFMAN true BASIS 6-31+G(d) GEN_SCFMAN true SCF_ALGORITHM diis SCF_CONVERGENCE 8 THRESH 14 SYMMETRY false SYM_IGNORE true SCF_FINAL_PRINT 1 FRGM_METHOD stoll SCFMI_MODE 1 !nonortho gen scfmi CHILD_MP true CHILD_MP_ORDERS 122 !DQ FD_MAT_VEC_PROD false $end

The frozen interaction energy in ALMO-EDA is defined as the energy difference between the unrelaxed frozen (Heitler-London) wave function and the isolated fragments. In other literature (*e.g.* Ref. Hopffgarten:2012), this interaction is often decomposed in a classical fashion:

(12.5) |

where the contribution from permanent electrostatics is defined as the Coulomb interaction between isolated fragment charge distributions:

(12.6) |

and the remainder constitutes the Pauli (or exchange) term. Such a decomposition (referred to as the classical decomposition below) is associated with two issues: (i) the evaluation of permanent electrostatics makes use of the “promolecule" state (whose density is the simple sum of monomer densities) rather than a properly anti-symmetrized wave function; (ii) when dispersion-corrected density functionals are utilized, the Pauli term contains dispersion interaction and thus loses its original meaning.

Horn *et al.* proposed a new scheme to further decompose the frozen term into contributions from permanent electrostatics (ELEC), Pauli repulsion (PAULI) and dispersion (DISP) [841]:

(12.7) |

This approach is compatible with the use of all kinds of density functionals except double-hybrids, and all three components of the FRZ term are computed with the antisymmetrized frozen wave function. The key step of this method is the orthogonal decomposition of the 1PDM associated with the frozen wave function into contributions from individual fragments: . This is achieved by minimizing an objective function as follows:

(12.8) |

while interfragment orthogonality is enforced between ’s. The readers are referred to Ref. Horn:2016b for more details about the orthogonal decomposition.

The ELEC term is then defined as the Coulomb interaction between distorted fragment densities ():

(12.9) |

The DISP term is evaluated by subtracting the dispersion-free part of the total exchange-correlation (XC) interaction, where an auxiliary “dispersion-free" (DF) XC functional is used in company with the primary XC functional:

(12.10) |

It is suggested that HF is an appropriate DFXC to be used for dispersion-corrected hybrid functionals (*e.g.* B97M-V, B3LYP-D3), while revPBE is appropriate for semi-local functionals (*e.g.* B97M-V).

The remainder of the frozen interaction goes into the PAULI term, which includes the net repulsive interaction given by eq. and the “dispersion-free" part of the XC interaction:

(12.11) |

The PAULI term and the ELEC term can also be combined together and reported as the dispersion-free frozen (DFFRZ) term if desired.

In Q-Chem’s implementation of “EDA2", the classical frozen decomposition and the new scheme defined by eqs. – are both calculated by default. The classical ELEC term only depends on monomer properties and the distances between fragments, therefore, it can be particularly useful for scenarios such as force field development (as the reference for permanent electrostatics). When the DISP term calculated by the new scheme is available, a modified classical Pauli term [868] is also reported, which is simply defined as

(12.12) |

*i.e.*, the dispersion contribution is removed from the classical Pauli term computed using its original definition. Alternatively, this can also be achieved without performing the orthogonal decomposition, by setting CLS_DISP to TRUE. This also evaluates the DISP term via eq. except that undistorted monomer densities (’s) are used instead of their distorted counterparts (’s).

FRZ_ORTHO_DECOMP

Perform the decomposition of frozen interaction energy based on the orthogonal decomposition of the 1PDM associated with the frozen wave function.

TYPE:

BOOLEAN

DEFAULT:

FALSE (automatically set to TRUE by EDA2 options 1–5)

OPTIONS:

FALSE

Do not perform the orthogonal decomposition.

TRUE

Perform the frozen energy decomposition using orthogonal fragment densities.

RECOMMENDATION:

Use default value automatically set by “EDA2".

**Note: **Users are allowed to turn off the orthogonal decomposition by setting FRZ_ORTHO_DECOMP to -1. Also, for calculations that involve ECPs, it is automatically set to FALSE since unreasonable results will be produced otherwise.

FRZ_ORTHO_DECOMP_CONV

Convergence criterion for the minimization problem that gives the orthogonal fragment densities.

TYPE:

INTEGER

DEFAULT:

6

OPTIONS:

RECOMMENDATION:

Use the default unless tighter convergence is preferred.

EDA_CLS_ELEC

Perform the classical decomposition of the frozen term.

TYPE:

BOOLEAN

DEFAULT:

FALSE (automatically set to TRUE by EDA2 options 1–5)

OPTIONS:

FALSE

Do not compute the classical ELEC and PAULI terms.

TRUE

Perform the classical decomposition.

RECOMMENDATION:

TRUE

EDA_CLS_DISP

Compute the DISP contribution without performing the orthogonal decomposition, which will then be subtracted from the classical PAULI term.

TYPE:

BOOLEAN

DEFAULT:

FALSE

OPTIONS:

FALSE

Use the DISP term computed with orthogonal decomposition (if available).

TRUE

Use the DISP term computed using undistorted monomer densities.

RECOMMENDATION:

Set it to TRUE when orthogonal decomposition is not performed.

DISP_FREE_X

Specify the employed “dispersion-free" exchange functional.

TYPE:

STRING

DEFAULT:

HF

OPTIONS:

Exchange functionals (

e.g.revPBE) or exchange-correlation functionals (e.g.B3LYP)supported by Q-Chem.

RECOMMENDATION:

HF is recommended for hybrid (primary) functionals (

e.g.B97X-V) and revPBE for semi-local ones (e.g.B97M-V). Other reasonable options (e.g.B3LYP for B3LYP-D3) can also be applied.

DISP_FREE_C

Specify the employed “dispersion-free" correlation functional.

TYPE:

STRING

DEFAULT:

NONE

OPTIONS:

Correlation functionals supported by Q-Chem.

RECOMMENDATION:

Put the appropriate correlation functional paired with the chosen exchange functional (

e.g.put PBE if DISP_FREE_X is revPBE); put NONE if DISP_FREE_X is set to an exchange-correlation functional.

The use of the FERF model for the evaluation of polarization energy and the further decomposition of the frozen term define the second generation of the ALMO-EDA method. Meanwhile, under the same code structure, the original AO-block based ALMO model and other related methods (such as the constrained relaxation of the frozen wave function [840] which renders the frozen energy variationally computed, and the polMO model [869] that arguably gives a lower limit to the polarization contribution) are also available. This entire set of methods implemented in Q-Chem based on GEN_SCFMAN (see Section 4.7) is termed as “EDA2". Up till today, it has covered most of the functionality of the original ALMO-EDA implementation in Q-Chem except the features related to CTA and COVP analysis. We look forward to switching to “EDA2" entirely in the near future.

The job control for EDA2 is largely simplified by a series of preset options provided by the developers. The option number is set through the EDA2 *$rem* variable (introduced below). Besides that, for the sake of flexibility, users are allowed to overwrite the values of part of the preset *$rem* variables:

Related to the isolated fragment calculations:

EDA_CHILD_SUPER_BASIS: use the super-system basis for fragment calculations (default: FALSE).

FRAGMO_GUESS_MODE: as introduced in Section 12.3 (default: 0).

Related to the decomposition of the FRZ term:

FRZ_ORTHO_DECOMP: it can be turned off by setting its value to -1 in the

*$rem*section

(default: TRUE).FRZ_ORTHO_DECOMP_CONV: as introduced in Section 12.7.3 (default: 6).

EDA_CLS_DISP: as introduced in Section 12.7.3 (default: FALSE).

DISP_FREE_X: as introduced in Section 12.7.3 (default: HF).

DISP_FREE_C: as introduced in Section 12.7.3 (default: NONE).

Related to the evaluation of POL:

Related to the evaluation of CT and BSSE:

EDA_NO_CT: skip the evaluation of the CT term in the EDA procedure

(default: FALSE (automatically turned on when SCFMI_FREEZE_SS = TRUE)).EDA_BSSE: use counterpoise-corrected monomer calculations to evaluate the BSSE

(default: FALSE).

EDA2

Switch on EDA2 and specify the option set number.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0

Do not run through EDA2.

1

Frozen energy decomposition + nDQ-FERF polarization

(the standard EDA2 option)

2

Frozen energy decomposition + (AO-block-based) ALMO polarization

(old scheme with the addition of frozen decomposition)

3

Frozen energy decomposition + oDQ-FERF polarization

(NOT commonly used)

4

Frozen wave function relaxation + Frozen energy decomposition + nDQ-FERF polarization

(NOT commonly used)

5

Frozen energy decomposition + polMO polarization

(NOT commonly used).

10

No preset. Completely controlled by user’s

$reminput(for developers only)

RECOMMENDATION:

Turn on EDA2 for Q-Chem’s ALMO-EDA jobs unless CTA with the old scheme is desired. Option 1 is recommended in general, especially when substantially large basis sets are employed. The original ALMO scheme (option 2) can be used when the employed basis set is of small or medium size (arguably no larger than augmented triple-). The other options are rarely used for routine applications.

**Example 12.311** Energy decomposition analysis for the ammonia-borane complex. The FERF-nDQ model is used for the POL term (as very large basis set is employed here), and decomposition of the frozen interaction energy is performed (Hartree-Fock is employed as the DFXC functional by default).

$molecule 0 1 -- 0 1 N 0.000000 0.000000 -0.727325 H 0.947371 0.000000 -1.091577 H -0.473685 -0.820448 -1.091577 H -0.473685 0.820448 -1.091577 -- 0 1 B 0.000000 0.000000 0.930725 H -1.165774 0.000000 1.243063 H 0.582887 -1.009590 1.243063 H 0.582887 1.009590 1.243063 $end $rem JOBTYPE eda EDA2 1 METHOD wB97M-V BASIS def2-QZVPPD SYMMETRY false MEM_TOTAL 4000 MEM_STATIC 1000 THRESH 14 SCF_CONVERGENCE 8 XC_GRID 000099000590 NL_GRID 1 FD_MAT_VEC_PROD false $end

**Example 12.312** Energy decomposition analysis of the water dimer with a low-cost model chemistry. The original ALMO model is used for the evaluation of polarization energy, and revPBE is chosen as the DFXC functional. Counterpoise correction for the BSSE is applied.

$molecule 0 1 -- 0 1 H1 O1 H1 0.95641 H2 O1 0.96500 H1 104.77306 -- 0 1 O2 H2 dist O1 171.85474 H1 180.000 H3 O2 0.95822 H2 111.79807 O1 -58.587 H4 O2 0.95822 H2 111.79807 O1 58.587 dist = 2.0 $end $rem JOBTYPE eda EDA2 2 METHOD b97m-v BASIS def2-svpd SCF_CONVERGENCE 8 THRESH 14 SYMMETRY false DISP_FREE_X revPBE DISP_FREE_C PBE EDA_BSSE true $end