Omega representation

Duo traditionally works in a Hund’s case (a) electronic basis, |\mathrm{state},\Lambda,S,\Sigma\rangle, i.e. the Lambda–S (or vib) contraction. Duo can now also work in an \Omega-based contracted representation, where the electronic + spin–orbit problem is diagonalised at each bond length and the resulting \Omega-labelled channels are used for vibrational contraction and for the final rovibronic basis.

The \Lambda S \rightarrow \Omega transformation (often called the state-interacting method) has been widely used to simplify the treatment of spin–orbit coupling in rovibronic calculations; see, e.g., :cite:p:21BeJoLia.SO`, [3], [4]. The key idea is to diagonalise the electronic Hamiltonian together with the Breit–Pauli spin–orbit Hamiltonian to obtain effective potentials for each spin–orbit component. While this can make the problem look “single-state-like”, strict equivalence with the original \Lambda S formulation requires transforming the full nuclear-motion Hamiltonian, which introduces spin–orbit-induced non-adiabatic couplings (NACs); see, e.g., [5], [6], [7].

Transforming to the \Omega representation

A general workflow for building the \Omega representation is:

  1. Solve the electronic Schrödinger equation to obtain electronic wavefunctions, and construct potential energy curves and spin–orbit coupling curves for the electronic states of interest.

  2. Build, at each bond length r, the electronic + spin–orbit Hamiltonian matrix \mathbf{H}_\Omega(r) = \mathbf{V}(r) + \mathbf{H}_{\rm SO}(r) in the chosen electronic basis.

  3. Diagonalise \mathbf{H}_\Omega(r) at each r to obtain spin–orbit-decoupled channels and effective potentials labelled by \Omega.

  4. To achieve exact equivalence with the original \Lambda S representation, apply the same r-dependent unitary transformation to the remaining parts of the rovibronic Hamiltonian. In particular, transforming radial derivative operators generates NAC terms that must be included for accurate energies, wavefunctions, and intensities.

The diatom in the \Lambda S and \Omega representations

In a Hund’s case (a) basis, the coupled rovibronic Schrödinger equation can be written as (see also the standard Duo theory in the manual):

(1)\left[ \frac{\hbar^2}{2\mu}\left(-\frac{d^2}{dr^2} + \frac{1}{r^2}\hat{\mathbf{R}}^2\right) + \mathbf{V}(r) + \mathbf{H}_{\rm SO}(r) \right]\vec{\chi}(r) = E_i\,\vec{\chi}(r),

where \mu is the reduced mass, r is the internuclear separation, \hat{\mathbf{R}} is the nuclear rotational angular momentum operator, \mathbf{V} contains diagonal Born–Oppenheimer PECs [8], and \mathbf{H}_{\rm SO} contains spin–orbit matrix elements (e.g. from ab initio electronic-structure calculations).

State-interacting diagonalisation

The state-interacting \Omega transformation is defined by diagonalising \mathbf{V}+\mathbf{H}_{\rm SO} at each r:

(2)\mathbf{U}^\dagger(r)\left(\mathbf{V}(r)+\mathbf{H}_{\rm SO}(r)\right)\mathbf{U}(r) = \mathbf{V}_\Omega(r),

where \mathbf{U}(r) is an r-dependent unitary matrix and \mathbf{V}_\Omega(r) is diagonal. This defines the transformation of the electronic basis

(3)|\mathrm{state},\Lambda,S,\Sigma\rangle \;\rightarrow\; |\mathrm{state},\Omega\rangle.

It is tempting to assume that \mathbf{V}_\Omega(r) yields fully decoupled single-channel rovibronic problems. However, to remain formally consistent, the kinetic-energy operators in Eq. (1) must also be transformed. The transformation of radial derivatives introduces NAC terms.

Spin–orbit-induced NACs from the vibrational kinetic energy

Upon transforming the radial kinetic-energy operator, one obtains additional terms of the standard adiabatic/NAC form (see diabatisation theory, e.g. [9], [10], [6], [7]):

(4)-\frac{\hbar^2}{2\mu}\mathbf{U}^\dagger \frac{d^2}{dr^2}\mathbf{U} = -\frac{\hbar^2}{2\mu}\left[ \frac{d^2}{dr^2} + \mathbf{W}^2 -\left(\frac{d}{dr}\mathbf{W} - \mathbf{W}\frac{d}{dr}\right) \right],

where

\mathbf{W}(r) = \mathbf{U}(r)\,\frac{d\mathbf{U}^\dagger(r)}{dr}

is a skew-Hermitian matrix of derivative couplings. The diagonal elements of \mathbf{W}^2 act as additional diagonal corrections (analogous in spirit to DBOC-like terms), while off-diagonal terms mediate non-adiabatic transitions between channels of the same \Omega.

Note

In the \Omega representation, “simplifying” the potential by diagonalisation relocates the physics into induced non-adiabatic terms. Neglecting these terms can lead to substantial errors in energies and wavefunctions, and therefore also in intensities and lifetimes (see, e.g. [], [6], [11]).

How to use the Omega representation in Duo

Switching from the standard Lambda–S contraction to the \Omega contraction requires only a change in the contraction block:

contraction
  omega
  vmax 20 40 40
end

Here the usual vib (Lambda–S) option is replaced by omega.

High-level algorithm used by Duo (omega mode)

When omega is selected, Duo performs (conceptually) the following steps:

  1. At each radial grid point r, build an extended electronic matrix and diagonalise it to obtain \mathbf{U}(r) and \mathbf{V}_\Omega(r). Duo then uses \mathbf{U}(r) to transform relevant curve-based operators to the \Omega representation and to generate the induced NAC terms arising from the r-dependence of \mathbf{U}(r).

    Note

    The “extended matrix” used for diagonalisation includes spin–orbit together with other J-independent electronic terms available in the model. Any terms not included at this stage are still handled in the full rovibronic Hamiltonian, but the precise partitioning of terms between “diagonalisation” and “final Hamiltonian” affects the structure of the induced NACs and should be treated consistently.

  2. Solve the pure vibrational problems for each \Omega channel using the chosen DVR method (e.g. sinc DVR), obtaining vibrational functions |\Omega, v\rangle.

  3. Use these vibrational functions to compute vibrational matrix elements of all required operators and curves (PECs, DMCs, couplings) in the \Omega representation.

  4. Construct the final rovibronic basis and Hamiltonian. Schematically,

    \Phi_{J,\Omega,v} \;=\; |J,\Omega\rangle\,|\Omega,v\rangle,

    which are then symmetrised (Wang functions) and used to build and diagonalise the rovibronic Hamiltonian.

  5. If requested, compute rovibronic intensities and/or line lists using the same workflows as in the Lambda–S mode.

Omega-specific output

The output is similar to the standard Lambda–S mode, but includes additional diagnostics for curves and operators transformed to the \Omega representation. For example, Duo can print PECs labelled by \Omega:

PECS in the Omega representation
#  #    Omega Lambda  Sigma  State
1  1     -1.0    0     -1.0 a3Sigma-
2  1      0.0    0      0.0 X1Sig+
...

and transformed multi-component operators such as the electronic spin operator (schematically shown here):

S = (S+,S-) in the Omega representation
#   i Omega State Lambda Sigma <-> j Omega State Lambda Sigma
1   1  -1.0   2    0    -1.0      1   0.0   1    0     0.0
...

Similarly, dipole moment curves can be printed in the \Omega representation:

Dipole in the Omega representation
#    Omega State Lambda Sigma   Omega  State Lambda Sigma
1      0.0   1    0     0.0       0.0   1    0     0.0
...

Warning and words of caution

Warning

The \Omega representation in Duo is currently a work in progress. It has been tested on a limited set of systems and couplings (see [11]) and may produce incorrect results for models involving additional couplings or corrections not yet fully validated in omega mode. If you use omega mode beyond the tested scope, it is strongly recommended to validate against the standard Lambda–S calculation.

Even when technically available, the fully decoupled (single-state) \Omega approximation should not be used blindly. Transforming to the \Omega representation unavoidably introduces induced NAC terms which may be essential for accurate energies and transition properties, especially for spin-forbidden bands. The broader lesson is that unitary “simplifications” must be accompanied by an accounting of the physics moved elsewhere in the Hamiltonian (see discussion and examples in [11]).

Keywords

omega

Option in the contraction block that requests contraction and rovibronic calculations in the \Omega representation. This triggers an r-dependent diagonalisation to define \Omega channels and includes the induced non-adiabatic couplings required for formal consistency.

Example:

contraction
  omega
  vmax 20 40 40
end
vib

Standard option in the contraction block for the Hund’s case (a) Lambda–S representation (the default behaviour in Duo). In this manual it may also be referred to as the Lambda–S contraction.

Example:

contraction
  vib
  vmax 20 40 40
end

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