10.2 Wave Function Analysis 10.2.8 Intrinsic Atomic Orbitals 10.2.10 Atomic dipoles and quadrupoles

10.2.9 Broken Bond Orbitals

(May 21, 2025)

Broken bond orbitals (BBOs) are a set of complete active space (CAS) orbitals developed by Sterling et al ¹²⁵³ Sterling A. J. et al.
J. Am. Chem. Soc.
(2024), 146, pp. 9532. Link to investigate the importance of orbital contraction on the formation of chemical bonds. BBOs are an set of uncontracted orbitals that span the space of isolated atoms/fragments that are then used to evaluate properties of a bonded system, effectively prohibiting orbital contraction effects (and some other orbital relaxation effects). As a result, the total (and kinetic) energy of the system can be decomposed into a sum of bonding effects that result from the total energy lowering that is possible using unrelaxed orbitals ( $\Delta E_{\mathrm{BBO}}$ ), and the further energy lowering that accompanies orbital relaxation ( $\Delta E_{\mathrm{rlx}}$ ):

\Delta E(r)=\Delta E_{\mathrm{BBO}}(r)+\Delta E_{\mathrm{rlx}}(r)

(10.25)

BBOs are generated according to the following procedure: (1) Obtain a set of localized orbitals $\{\phi\}$ at the CASSCF (complete active space self-consistent field) level by choosing a bond length at which the bond is unambiguously broken – chosen to be 10 Å for uncharged fragments, and 1000 Å for charged fragments; (2) These BBOs are then used as a basis for a subsequent CASCI (complete active space configuration interaction) calculation, by translating the BBOs to the desired bond length. Since these initial BBOs are no longer orthogonal upon translation, a judiciously-chosen orthogonalization procedure was chosen that first symmetrically orthogonalizes all inactive (core) orbitals:

\mathbf{\tilde{C}}_{\mathrm{c}}=\mathbf{C}_{\mathrm{c}}\mathbf{S}_{\mathrm{c}}% ^{-\frac{1}{2}}

(10.26)

followed by projection of the orthogonalized core out of the active (valence) orbitals, and subsequent renormalization and symmetric orthogonalization of these projected valence orbitals:

\mathbf{C}_{\mathrm{v,proj}}=(\mathbf{I-\tilde{P}_{\mathrm{c}}}\mathbf{S}_{\mu% \nu})\mathbf{C}_{\mathrm{v}}

(10.27)

where $\mathbf{C}$ is the coefficient matrix, $\mathbf{S}$ is the overlap matrix, $\mathbf{P}=\mathbf{CC^{T}}$ , subscripted c, v denote core and valence spaces, respectively, and $\mu,\nu$ , denote atomic orbitals.

This approach ensures qualitatively correct bond dissociation, and enables evaluation of total and kinetic energy contributions to bond formation in the absence of orbital relaxation.

To run a BBO calculation, link a pair of CAS jobs, where the first job uses a CASSCF procedure to generate the set of BBOs, followed by a CASCI job that reads in the generated BBOs and calculates the energy at the desired bond length. Ensure that the SCF procedure in the second job is skipped to avoid unwanted modification of the BBOs. These BBOs can be visualized by specifying GUI = 2 in the $rem block, and their natural occupation numbers and kinetic energies are printed in the output file by default.

Example 10.5 Input for a BBO calculation on Li ${}_{2}$ .

$molecule
0 1
Li
Li 1 R

R = 10.0  ! choose a long bond length to generate BBOs
$end

$rem
  JOBTYPE                    sp
  gen_scfman                 TRUE
  EXCHANGE                   HF
  BASIS                      cc-pVDZ
  SCF_GUESS                  SAD
  SCF_ALGORITHM              GDM
  MAX_SCF_CYCLES             250
  CAS_METHOD                 2     !1 for CAS-CI, 2 for CASSCF
  CAS_M_S                    0     !M_s value*2
  CAS_N_ELEC                 2     !N_elec
  CAS_N_ORB                  2     !N_orb
  CAS_N_ROOTS                1     !N_roots
  CAS_SOLVER                 0     !2=ASCI, 1=Olsen, 0=naive
  INTEGRAL_SYMMETRY          false
  POINT_GROUP_SYMMETRY       false
$end

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10.2.9 Broken Bond Orbitals