x1fd3

A program to deal with the adiabatic $X^1\Sigma^+$ state of diatomic molecules.

Requirements

• python >= 3.10
• numpy
• scipy
• tkinter
• matplotlib

Test

On *nix systems run:

test/test.sh test

for integration testing. Zero output is OK. Note that the first and last tests ("PecApprox_1.log" vs "test/PecApprox.ref" and "FitExp_1.log" vs "test/FitExp.ref") could fail since they are slightly system-dependent. If "PEC fit done" is in "PecApprox_1.log" and "FitExp_1.log" they works just fine.

For unit testing run:

python test/unittests.py

Usage

python -m x1fd3 [mode] [input files]

Available modes:

• GUI - TK wrapper for all other modes
• PecApprox - Potential energy curve (PEC) approximation with analytic function
• LevelsPW - Vibrational-rotational level calculation for given point-wise PEC
• LevelsAn - Vibrational-rotational level calculation for given analytic PEC
• SpectrumPW - Vibrational-rotational spectrum calculation for given point-wise PEC and point-wise dipole moment (DM)
• SpectrumAn - Vibrational-rotational spectrum calculation for given analytic PEC and point-wise DM
• FitExp - fit analytic PEC to reproduce given experimental vibrational-rotational levels

No input files required to run GUI. For CLI-based modes 2, 3, or 4 files should to be provided. Run any of these modes with no arguments to see help.

Mode details

PecApprox

Point-wise PEC is approximated with Extended Morse Oscillator (EMO) function:

$$U_{EMO} = D_e \left[ 1 - e^{-\beta(R) \cdot (R-R_e)} \right]^2$$

where

$$\beta(R) = \sum_{i=0}^N \beta_i \cdot y^i$$

and

$$y = \frac{R^q - R_{ref}^q}{R^q + R_{ref}^q}$$

$D_e$, $R_e$ and $\beta_0$, ..., $\beta_N$ are optimized within non-linear least squares.

Note 1: MLR and DELR are available as well.

Note 2: By default $U(R_e) = 0$ is assumed. To change it one can set $T_e \equiv U(R_e)$ or $T_d \equiv U(+\infty)$.

LevelsPW

Vibrational-rotational levels are found by solving the radial Schrodinger equation:

$$\left[-\frac{\hbar^2}{2\mu}\frac{d2}{dR^2} + U(R) + \frac{\hbar^2 J(J+1)}{2\mu R^2}\right]\Psi_v = E\Psi_v$$

The finite-difference scheme is used within the scheme 3-point approximation of 2nd derivative:

$$f''(x_i) \approx \frac{ f(x_{i-1}) - 2f(x_i) + f(x_{i+1}) }{ \Delta x^2}$$

Within this method the original differential equation is transformed to tridiagonal symmetric matrix eigenvalue problem.

PEC ($U(R)$) is required in point-wise format, spline interpolation is used.

LevelsAn

Similar to LevelsPW, but with analytic (EMO, etc) representation of PEC.

SpectrumPW

LevelsPW with additional matrix elements calculation for given point-wise DM ($D(R)$) and lower and upper vibrational quantum numbers $v_1$ and $v_2$:

$$\mu = <\Psi_{v1}|D|\Psi_{v2}>$$

Total transition matrix elements for absorption and emission spectra can be estimated as:

$$M^2 = \mu^2 \cdot S$$

where $S$ are Hoenl-London factors.

SpectrumAn

Similar to SpectrumPW, but with analytic (EMO, etc) representation of PEC.

FitExp

PEC is optimized to fit given set of vibrational-rotational experimental levels. Least-squares procedure is based on Hellmann–Feynman theorem:

$$\frac{\partial E}{\partial p_i} = \left< \Psi_v \middle | \frac{\partial U(p_1,p_2,...) }{\partial p_i} \middle | \Psi_v \right >$$

where $p_i$ are $D_e$, $R_e$ and $\beta_0$, ..., $\beta_N$.