10.2 Wave Function Analysis 10.2.6 Oxidation State Localized Orbitals 10.2.8 Basic Excited-State Analysis of CIS and TDDFT Wave Functions

10.2.7 Intrinsic Atomic Orbitals

(April 13, 2024)

Due to the form of commonly used basis sets, SCF wave functions $|\Phi\rangle$ are often not easily interpretable in terms of their atomic orbital (AO) basis functions. Especially for large basis sets it is not possible to achieve an unambiguous association of a basis function’s contribution to the wave function with the atom it is centered on. An expansion of the molecular orbitals over a minimal basis (of free-atom AOs), on the other hand, does allow for a simple interpretation of the wave function. However, free-atom atomic orbitals cannot yield a qualitatively correct minimal basis expansion as these AOs do not account for the polarization due to the molecular environment. The approach by Knizia overcomes this drawback by determining a set of polarized AOs $\{|\rho\rangle\}$ , termed intrinsic atomic orbitals (IAOs) ⁶³⁰ Knizia G.
J. Chem. Theory Comput.
(2013), 9, pp. 4834. Link , which exactly express the occupied space $\mathcal{O}=\sum_{i}|i\rangle\langle i|$ of the wave function $|\Phi\rangle$ . As a consequence, atomic properties, such as partial charges, become accessible through IAOs. The IAO-derived atomic charges of a closed-shell system are then given by

q_{A}=Z_{A}-\sum_{\rho\in A}\langle\rho|\gamma|\rho\rangle,

(10.11)

where $Z_{A}$ is the nuclear charge of atom $A$ and $\gamma$ represents the closed-shell SCF density matrix $\gamma=2\sum_{i}|i\rangle\langle i|$ .

The resulting IAOs in conjunction with an orbital localization scheme can be used to construct bonding orbitals, termed intrinsic bond orbitals (IBOs). The applied localization scheme is similar to the Pipek-Mezey approach ⁹⁸¹ Pipek J., Mezey P. G.
J. Chem. Phys.
(1989), 90, pp. 4916. Link and effectively minimizes the number of atoms on which an orbital is centered by maximizing the functional

L=\sum_{A}^{\text{atoms}}\sum_{i^{\prime}}^{\text{occ}}[n_{A}(i^{\prime})]^{4},

(10.12)

with respect to unitary orbital rotations. In the above equation, $n_{A}(i^{\prime})$ is the number of electrons from rotated orbital $|i^{\prime}\rangle$ located on the IAOs of atom A:

n_{A}(i^{\prime})=2\sum_{\rho\in A}\langle i^{\prime}|\rho\rangle\langle\rho|i% ^{\prime}\rangle.

(10.13)

Intrinsic bond orbitals exactly represent the occupied molecular orbitals and can thus be used to interpret the electronic molecular structure.

Calculation of intrinsic bond orbitals is controlled by the following $rem variable:

DO_IBO

DO_IBO
       Enables IBO procedure
TYPE:
       BOOL
DEFAULT:
       FALSE
OPTIONS:
       FALSE Do not calculate IBOs TRUE Run the IBO procedure
RECOMMENDATION:
       None

Additional control parameters for the IBO procedure can be declared in the $loco input section and are listed below.

MIN_BASIS
       Specifies the free atom basis used for the construction of IAOs.
INPUT SECTION: $loco
TYPE:
       STRING
DEFAULT:
       MINAO
OPTIONS:
       MINAO Use truncated cc-pVTZ basis (same as in original publication) STO-3G Use STO-3G basis STO-6G Use STO-6G basis autoSAD Construct minimal basis from autoSAD procedure.
RECOMMENDATION:
       Use autoSAD procedure.

EXP_FOUR
       Whether to use an exponent of $p=4$ in the localization functional (Eq. 10.12).
INPUT SECTION: $loco
TYPE:
       BOOL
DEFAULT:
       TRUE
OPTIONS:
       TRUE Use exponent $p=4$ (same as in original publication) FALSE Use exponent $p=2$
RECOMMENDATION:
       An exponent of $p=4$ is preferred as it leads to discrete localizations in aromatic systems ( $p=2$ does not).

IBO_POP_THRESH
       Threshold for printing IBO occupations.
INPUT SECTION: $loco
TYPE:
       FLOAT
DEFAULT:
       0.1
OPTIONS:
       0.1 Prints IBO occupations $>10\%$
RECOMMENDATION:
       Use the default.

IBO_MEM
       Memory (in MB) for IBO procedure
INPUT SECTION: $loco
TYPE:
       INTEGER
DEFAULT:
       500
OPTIONS:

RECOMMENDATION:
       500MB should suffice for most purposes. Increase only when needed.

EXPORT_MOLDEN
       Whether to write IBOs in Molden format.
INPUT SECTION: $loco
TYPE:
       BOOL
DEFAULT:
       FALSE
OPTIONS:
       TRUE Export IBOs in Molden format (ibo.molden). FALSE Do not create Molden file.
RECOMMENDATION:
       None

Example 10.3 Input for calculating atomic partial charges and bond populations via the IAO/IBO procedure.

$molecule
0 1
O        0.0000000000     0.0000000000     0.1172309766
H       -0.7626482594     0.0000000000    -0.4685977726
H        0.7626482594     0.0000000000    -0.4685977726
$end

$rem
method          = wB97M-V
basis           = def2-TZVP
mem_total       = 4000
mem_static      = 500
scf_convergence = 9
thresh          = 14
gui             = 2
do_ibo          = true
$end


$loco
min_basis     = autosad
export_molden = true
$end

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10.2.7 Intrinsic Atomic Orbitals