Treating unbound states¶

Duo is primarily designed for bound-state rovibronic problems, using an effective boundary condition that wavefunctions vanish at the borders of the radial grid ( $r_{\rm min}$ and $r_{\rm max}$ ). In practice, however, many diatomic models include electronic states with dissociation limits inside the energy range of interest, or with continua above dissociation. In such cases Duo may produce solutions that are formally “eigenstates” on the finite grid but correspond to continuum-like (unbound) behaviour. These states can contaminate spectra and line lists if not treated carefully.

This section describes two practical criteria implemented in Duo to identify (un)bound character and how to use them to include or exclude states in intensity/line-list calculations. It also explains how to separate bound–continuum and continuum–bound contributions in intensity calculations.

Identifying unbound states using a boundary density test¶

A continuum-like (unbound) wavefunction typically carries non-negligible probability density close to the outer grid boundary. Duo can therefore flag a state $\psi_\lambda(r)$ as unbound by integrating the density in a small region of width $\delta$ at the right boundary:

$\epsilon = \int_{r_{\rm max}-\delta}^{r_{\rm max}} \left|\psi_\lambda(r)\right|^2 \, dr.$

If $\epsilon$ exceeds a threshold $\epsilon_{\rm thr}$ , the state is treated as unbound:

$\epsilon > \epsilon_{\rm thr}.$

The threshold $\epsilon_{\rm thr}$ is controlled by the input keyword thresh_bound. The integration width $\delta$ (in Å) is controlled by thresh_delta_r.

Typical defaults are:

$\epsilon_{\rm thr} \sim 10^{-8}$ (thresh_bound)
$\delta = 0.5\,\AA$ (thresh_delta_r)

Note

This criterion depends on the choice of $r_{\rm max}$ and $\delta$ . For quasi-bound (resonance) states the separation between “bound” and “continuum-like” behaviour is not universal; the thresholds often need to be tuned for a given molecule and energy range.

Identifying unbound states using $\langle r \rangle$ ¶

A complementary indicator of continuum-like behaviour is the expectation value of the bond length:

$\langle r \rangle = \int_{r_{\rm min}}^{r_{\rm max}} \psi_\lambda(r)\, r \, \psi_\lambda^{*}(r)\, dr.$

Duo can flag a state as unbound if $\langle r \rangle$ exceeds a user-defined threshold $r_{\rm thresh}$ . This threshold is specified via thresh_bound_rmax.

Note

When both criteria are enabled, to be a bound state, it must satisfy both criteria, for $\langle r \rangle < r_{\rm thresh}$ and $\epsilon < \epsilon_{\rm thr}$ . That is, a state is unbound if any of these two conditions is satisfied.

The .states file can include the bound/unbound label (b/u) and the value of $\langle r \rangle$ used for classification (last column), e.g.

   0.000000      4     0.5 + e X2Pi         0  1    -0.5     0.5 b   1.145131
2652.362427      4     0.5 + e X2Pi         1  1    -0.5     0.5 b   1.188210
5200.319353      4     0.5 + e X2Pi         2  1    -0.5     0.5 b   1.230171
7642.107722      4     0.5 + e X2Pi         3  1    -0.5     0.5 b   1.275067
9977.930743      4     0.5 + e X2Pi         4  1    -0.5     0.5 b   1.314642
25367.675041      4     0.5 + e B2Sigma-     0  0     0.5     0.5 b   1.229628

Excluding unbound states (bound-bound spectra)¶

Once unbound states are identified, they can be excluded from intensity or line-list calculations by adding the keyword bound to the intensity section. This instructs Duo to compute bound–bound transitions only.

Example (using the density criterion):

intensity
  absorption
  bound
  thresh_intens   1e-50
  thresh_bound    1e-6
  thresh_delta_r  1.0
  temperature     3000.0
  linelist        AlCl-37_61_J160
  J               0, 20
  freq-window     0.0, 48000.0
  energy low      0.0, 30000.0, upper 0.0, 48000.0
end

Here, the integrated density $\epsilon$ over the region $[r_{\rm max}-\delta, r_{\rm max}]$ with $\delta=1\,\AA$ is compared to $\epsilon_{\rm thr}=10^{-6}$ . The state is considered unbound if $\epsilon>\epsilon_{\rm thr}$ .

Average-density option¶

Because $\epsilon$ scales with $\delta$ , it can be convenient to use the average density over the integration region:

$\bar{\epsilon} = \frac{\epsilon}{\delta}.$

This criterion is enabled using thresh_average_density, and the state is treated as unbound if $\bar{\epsilon} > \bar{\epsilon}_{\rm thr}$ .

Example:

intensity
  absorption
  bound
  thresh_bound            0.1
  thresh_delta_r          1.0
  thresh_average_density  1e-4
  thresh_intens           1e-40
  thresh_line             1e-40
  thresh_dipole           1e-7
  temperature             750.0
  linelist                AlH_446_A-X_L60.695_J10
  J                       0.0, 1.0
  freq-window             0.0, 30000.0
  energy low             -0.001, 30000.0, upper -0.0, 30000.0
end

The default value of $\bar{\epsilon}_{\rm thr}$ is typically $\sim 10^{-8}$ .

Selecting transitions involving unbound states¶

Duo can include transitions involving continuum-like states using the keyword unbound inside the intensity section. This is useful for modelling photodissociation continua, predissociation features, or any spectra where one side of a transition lies above dissociation.

By default, a single keyword unbound enables both of the following classes of transitions:

bound - unbound (unbound upper states), sometimes referred to as bound–continuum;
unbound - bound (unbound lower states), sometimes referred to as continuum–bound.

To process these two contributions independently, Duo accepts an optional second selector after unbound:

unbound upper: include only transitions whose upper state is classified as unbound;
unbound lower: include only transitions whose lower state is classified as unbound.

If the selector is omitted, unbound is equivalent to requesting both upper and lower contributions.

Examples¶

Bound - unbound (unbound upper states only):

intensity
  absorption
  unbound upper
  ...
end

Unbound - bound (unbound lower states only):

intensity
  absorption
  unbound lower
  ...
end

Both classes simultaneously:

intensity
  absorption
  unbound
  ...
end

Using $\langle r \rangle$ thresholds¶

The $\langle r \rangle$ criterion is enabled with thresh_bound_rmax:

intensity
  absorption
  unbound upper
  thresh_intens      1e-50
  thresh_bound       1e-6
  thresh_bound_rmax  3.0
  temperature        3000.0
  linelist           AlCl-37_61_J160
  J                  0, 20
  freq-window        0.0, 48000.0
  energy low         0.0, 30000.0, upper 0.0, 48000.0
end

State-dependent thresholds can be provided by listing one value per electronic state, in the same order as the states are defined (e.g. by the states keyword):

intensity
  absorption
  unbound upper
  ...
  thresh_bound_rmax  3.0  5.5  4.0
  ...
end

Note

When both criteria are requested, the $\langle r \rangle$ threshold can be used as a boundness safeguard: a state is considered bound if $\langle r \rangle < r_{\rm thresh}$ even if the boundary density test is marginal. A state is considered unbound if it is not bound.

Printing unbound diagnostics in the `.states` file¶

By default, unbound handling adds a bound/unbound label (b/u) to the .states file. When thresh_bound_rmax is used, the corresponding value of $\langle r \rangle$ is printed next to this label. The boundary-density integral $\epsilon$ can also be printed by adding print_bound_density:

intensity
  absorption
  unbound
  thresh_bound       1e-4
  thresh_delta_r     1.0
  thresh_bound_rmax  2.0
  print_bound_density
  temperature        3000.0
  linelist           CH
  J                  0.5, 10.5
  freq-window        0.0, 50000.0
  energy low        -0.001, 20000.0, upper -0.0, 50000.0
end

Example excerpt from the resulting .states file:

   0.000000      4     0.5 + e X2Pi         0  1    -0.5     0.5 b   1.13934 0.58E-21
2720.584833      4     0.5 + e X2Pi         1  1    -0.5     0.5 b   1.17893 0.17E-21
5314.682367      4     0.5 + e X2Pi         2  1    -0.5     0.5 b   1.22026 0.47E-21
7784.710908      4     0.5 + e X2Pi         3  1    -0.5     0.5 b   1.26339 0.97E-22
10132.366441      4     0.5 + e X2Pi         4  1    -0.5     0.5 b   1.30890 0.31E-21
25771.515741      4     0.5 + e B2Sigma-     0  0     0.5     0.5 b   1.21463 0.83E-21
27530.492176      4     0.5 + e B2Sigma-     1  0     0.5     0.5 b   1.32459 0.28E-19
27877.572537      4     0.5 + e B2Sigma-     2  0     0.5     0.5 u   4.15693 0.17E-01
27905.414170      4     0.5 + e B2Sigma-     3  0     0.5     0.5 u   4.96461 0.18
27947.636255      4     0.5 + e B2Sigma-     4  0     0.5     0.5 u   4.79088 0.28
 ...

Keywords¶

bound¶

Used inside the intensity block to compute bound–bound spectra/line lists only. Duo first classifies rovibronic states as bound/unbound using the criteria controlled by thresh_bound, thresh_delta_r, thresh_average_density, and/or thresh_bound_rmax, and then keeps only transitions whose lower and upper levels are classified as bound.

lower¶

Selector used only as a second keyword after unbound in the intensity block. unbound lower keeps transitions with unbound lower levels (unbound-bound).

print_bound_density¶

Requests printing of the boundary-density integral $\epsilon$ in the .states file (typically as an extra final column), alongside the bound/unbound label and (if requested) $\langle r \rangle$ .

thresh_average_density¶

Enables an average-density variant of the boundary criterion using

$\bar{\epsilon} = \epsilon/\delta.$

A state is treated as unbound if $\bar{\epsilon} > \bar{\epsilon}_{\rm thr}$ .

thresh_bound¶

Threshold $\epsilon_{\rm thr}$ used in the boundary-density criterion. Duo evaluates

$\epsilon = \int_{r_{\rm max}-\delta}^{r_{\rm max}} |\psi_\lambda(r)|^2 \, dr,$

where $\delta$ is controlled by thresh_delta_r. A state is treated as unbound if $\epsilon > \epsilon_{\rm thr}$ .

thresh_bound_rmax¶

Threshold $r_{\rm thresh}$ (in Å) used in the $\langle r \rangle$ criterion. Duo computes

$\langle r \rangle = \int_{r_{\rm min}}^{r_{\rm max}} \psi_\lambda(r)\, r\, \psi_\lambda^{*}(r)\, dr,$

and flags a state as unbound if $\langle r \rangle > r_{\rm thresh}$ .

The keyword can accept either a single value (applied to all states), or one value per electronic state (applied in the same order as the electronic states are defined in the input).

thresh_delta_r¶

Width $\delta$ (in Å) of the outer-boundary integration region used in the boundary-density criterion for thresh_bound (and, if enabled, for the average-density criterion thresh_average_density).

unbound¶

Used inside the intensity block to include transitions involving unbound states.

unbound (without a selector) enables both classes simultaneously: bound-unbound and unbound-bound.
unbound upper keeps only transitions whose upper level is classified as unbound (bound-unbound).
unbound lower keeps only transitions whose lower level is classified as unbound (unbound-bound).

Examples:

intensity
  absorption
  unbound upper
  ...
end

intensity
  absorption
  unbound lower
  ...
end

upper¶

Selector used only as a second keyword after unbound in the intensity block. unbound upper keeps transitions with unbound upper levels (bound-unbound).

Treating unbound states¶

Identifying unbound states using a boundary density test¶

Identifying unbound states using $\langle r \rangle$ ¶

Excluding unbound states (bound-bound spectra)¶

Average-density option¶

Selecting transitions involving unbound states¶

Examples¶

Using $\langle r \rangle$ thresholds¶

Printing unbound diagnostics in the `.states` file¶

Keywords¶

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Treating unbound states¶

Identifying unbound states using a boundary density test¶

Identifying unbound states using ¶

Excluding unbound states (bound-bound spectra)¶

Average-density option¶

Selecting transitions involving unbound states¶

Examples¶

Using thresholds¶

Printing unbound diagnostics in the .states file¶

Keywords¶

Identifying unbound states using $\langle r \rangle$ ¶

Using $\langle r \rangle$ thresholds¶

Printing unbound diagnostics in the `.states` file¶