Imaginary time dependent Schrodinger equation

In the diflfiision QMC (DMC) method [114. 119], the evolution of a trial wavefiinction (typically wavefiinctions of the Slater-Jastrow type, for example, obtained by VMC) proceeds in imaginary time, i = it, according to the time-dependent Schrodinger equation, which then becomes a drfifiision equation. All... 

This equation is exact and constitutes an iterative equation equivalent to the time-dependent Schrodinger equation [185,186]. The iterative process itself does not involve the imaginary number i therefore, if h(f) and )( — x) were the real parts of the wavepacket, then (f + x) would also be real and would be the real part of the exact wavepacket at time (f + x). Thus, if )( ) is complex, we can use Eq. (4.68) to propagate the real part of 4>(f) forward in time without reference to the imaginary part. 

If we fix a realization of the path Q(t), then, when performing the path integration over q, the particle may be treated as if it were subject to a time-dependent potential Vint[<2(r), q]. From traditional quantum mechanics it is clear that this integration is equivalent to the solution of the time-dependent Schrodinger equation in imaginary time ... 

The QMC method generates a many-body ground state wavefunction by propagating the time-dependent Schrodinger equation along the imaginary time axis. When the time variable is transformed as T = -it/h, the equation becomes iso-morphous to a diffusion equation that includes source and sink terms ... 

The easiest way to see this is simply to substitute a complex energy into the time-dependent Schrodinger equation the real part of the energy can then be associated with an oscillatory eigenfunction, while the imaginary part is associated with exponential decay. 

An improvement of the variational QMC can be obtained by the diffusion QMC approach. Consider the time-dependent Schrodinger equation, where the time is replaced with an imaginary time variable t = it. 

The diffusion Monte Carlo (DMC) method is based on the simulation of the diffusion equation. Let us start with the primitive DMC method. By replacing the real time t with the imaginary time r = it, the time-dependent Schrodinger equation is rewritten as... 

Because of its similarity to the time-dependent Schrodinger equation, Eq. [12] is often referred to as the Schrodinger equation in imaginary time. The analogy is formally correct, since solutions of the time-dependent Schrodinger equation have equivalent real and imaginary parts under steady state conditions. 

There is no way to obtain the time-dependent Schrodinger equation from a classical wave equation. The classical wave equation of a vibrating string, Eq. (14.3-3), is second order in time. It requires two initial conditions (an initial position and an initial velocity) to make a general solution apply to a specific case. The uncertainty principle of quantum mechanics (to be discussed later) implies that positions and velocities cannot be specified simultaneously with arbitrary accuracy. For this reason only one initial condition is possible, which requires the Schrodinger equation to be first order in time. The fact that the equation is first order in time also requires that the imaginary unit i must occur in the equation in order for oscillatory solutions to exist. 

A second, more accurate step, takes the optimised Jastrow factor as data and carries out a diffusion quantum Monte Carlo (DMC) calculation, based on transforming the time-dependent Schrodinger equation to a diffusion problem in imaginary time. An ensemble in configuration space is propagated to obtain a highly accurate ground state. 

If the diffusivity is imaginary, the diffusion equation has the same form as the time-dependent Schrodinger s equation at zero potential. Also, Eq. 4.18 implies that the velocity of the diffusant can be infinite. Schrodinger s equation violates this relativistic principle. 

We now turn to approaches that begin with an arbitrary initial function and can, in principle, be iterated to an exact or accurate solution of the SE. The earliest approach is Green s function Monte Carlo (GFMC) in which the time-independent Schrodinger equation is employed [24] DMC was developed later and follows from the time-dependent SE (TDSE) in imaginary time. 

A useful method that can be used in both time-independent and time-dependent calculations on chemical reactions is to add a negative imaginary potential (NIP) to the overall potential in carefully selected regions. This enables reactive flux into particular channels to be absorbed by the NIP and it is then not necessary to integrate the SchrOdinger equation further in that channel as the total reactive flux into it has been obtained. This is useful when state-to-state information is not needed but an overall reactive cross-section is required. The NIP also obviates the need for complicated coordinate systems in chemical reactions (such as hyperspherical coordinates) but the most appropriate parameters for the NIP have to be carefully tested. 

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