2023-02-01 | Reading time: 5 min
Welcome to my monthly blog series, where I share my ‘AHA’ moments in my research journey. In this first post, I want to take you on my journey of jumping into action and share with you some of my insights about the 📚 surface hopping method.
It all begins with the time-dependent electronic wave-function, expressed as a linear combination of the electronic basis states (1):
\[\displaystyle \Psi(R,t) = \sum_{a} c_{a}(t)\left|\varphi_{a}(R;t)\right\rangle\]In this formula, a runs over all basis states, ca are time-dependent coefficients, and \(\varphi_i(R;t)\) are the basis states. The latter case is, where things get interesting – the basis states are implicit time-dependent, as they depend on the coordinates of the nuclei (R), which are indeed time-dependent (This is indicated in the formula by the semicolon.)!
We start with the time-dependent Schrödinger equation (2):
\[\displaystyle i \frac{\partial \Psi(t)}{\partial t} = \hat{H} \Psi(t)\]When we insert (1) into the time-dependent Schrödinger equation (2), we get the following equation:
\[\displaystyle i \frac{\partial}{\partial t} \left( \sum_{a} c_{a} \left|\varphi_{a}\right\rangle \right) = \hat{H} \sum_{a} c_{a}\left|\varphi_{a} \right\rangle\]First we use the product rule for the derivative:
\[\displaystyle i \left( \sum_a \left|\varphi_{a}\right\rangle \frac{\partial}{\partial t}c_a + \sum_a c_a \frac{\partial}{\partial t} \left|\varphi_{a}\right\rangle \right) = \hat{H} \sum_{a} c_{a}\left|\varphi_{a} \right\rangle\]Next, we use the chain rule:
\[\displaystyle i \left( \sum_a \left|\varphi_{a}\right\rangle \frac{\partial}{\partial t}c_a + \sum_a c_a \frac{\partial R}{\partial t} \frac{\partial}{\partial R} \left|\varphi_{a}\right\rangle \right) = \hat{H} \sum_{a} c_{a}\left|\varphi_{a}\right\rangle\]Now we project on a state:
\[\displaystyle i \left( \frac{\partial}{\partial t}c_b + \sum_a c_a \frac{\partial R}{\partial t} \left\langle\varphi_b | \nabla_R\varphi_a\right\rangle \right) = \sum_a c_a \left\langle \varphi_b | \hat{H} | \varphi_a\right\rangle\]🔖 Note: In order to get the first derivative of the basis function, we have to use the chain rule, as the basis function is time dependent via the position of the nuclei R. By using the chain rule, we end up with a product of velocity and non-adiabatic couplings:
\[\displaystyle v = \frac{\partial R}{\partial t}\] \[\displaystyle C_{ba} = \left\langle\varphi_b | \nabla_R\varphi_a\right\rangle\]With
\[\displaystyle H_{ba} = \left\langle \varphi_b | \hat{H} | \varphi_a\right\rangle\]the general formula for the quantum amplitudes along the trajectory path, reads:
\[\displaystyle \frac{\partial}{\partial t} c_b = - \sum_a c_a \left(i H_{ba} + v C_{ba} \right)\]👩🔬 And there we go, my ‘AHA’ moment of January 2023. I hope you get inspired by my learning journey. I look forward to sharing my next ‘AHA’ moment with you in March!