# Units of derivative coupling (non-adiabatic coupling) in output

Dear all,
In the calculation of the non-adiabatic couplings, the (closely related) derivative coupling are written in the output (see below), but the units are not specified, nor did I find it in the manual. Does anyone know what units are used here?
Koen

Example output:
---------------------------------------------------
SF-CIS derivative coupling with ETF
Atom         X              Y              Z
---------------------------------------------------
1      -0.141357      -0.001891      -0.285072
2      -0.132861      -0.003380       0.293653
3      -0.009087       0.000272      -0.003756
4       0.137841      -0.038263       0.764793
5       0.267727       0.082422      -0.850842
6       0.006263       0.001799      -0.019702
7      -0.549606      -0.088453       0.543794
8       0.277761       0.044026      -0.148716
9      -0.057628      -0.055786       0.311969
10       0.009594      -0.000041       0.002089
11       0.295452      -0.000028       0.068386
12      -0.000077      -0.002123      -0.002412
13       0.039318       0.015193      -0.362853
14      -0.005108       0.001291      -0.008390
15      -0.281295       0.004393      -0.112621
16       0.001311       0.002530      -0.001512
17       0.174245       0.050989      -0.316830
18       0.009030      -0.000077      -0.004075
19      -0.073574      -0.022727       0.234137
20       0.016164       0.004927      -0.051426
21       0.015890       0.004928      -0.050614
---------------------------------------------------
Derivative coupling time:  CPU 189.32 s  wall 39.17 s

One way to see it is to take the exact derivative coupling expression, $\mathbf{d}{IJ} = \left<\Psi{I}|\frac{\partial}{\partial\mathbf{R}}|\Psi_{J}\right>$, and rewrite it in finite difference form. For a single Cartesian coordinate $x$ of the vector, this would be

$$\frac{\left<\Psi_{I}(x)|\Psi_{J}(x + \Delta x)\right> - \left<\Psi_{I}(x)|\Psi_{J}(x - \Delta x)\right>}{2\Delta x} + O[(\Delta x)^2]$$

The overlap between states is unitless, meaning the numerator has no units, but the denominator has units of length. This means the derivative coupling vector has dimensions of inverse length. Since the output is in atomic units, the final units are in inverse bohr.

I adapted this idea from https://doi.org/10.1063/1.4820485, specifically equation 52b.