86
3 Ab initio Electronic Structure for Few-Body Systems
In the VMC method, therefore, it is crucial the choice of the tentative T (r) that
determines the value of the observable computed by the simulation.
On the contrary, the DMC approach avoids the need for setting an approximate
tentative wavefunction by resorting to the use of the time-dependent formalism
H(r, τ ) = −
∂(r, τ )
∂τ
(3.4)
where τ is the imaginary time (τ = it). In Eq. 3.4 (r, τ ) tends to relax to the true
ground state wavefunction as τ → ∞ even when one begins with any arbitrary function at τ = 0 ((R, τ = 0)). In other words, the DMC trial wavefunction converges
in any case to the numerically accurate solution and its propagation is only a mathematical expedient when the algorithm is accurate and efficient. Accordingly, one
can drop τ from the notation when not strictly necessary. In addition, the method is
of particular interest for distributed computing because the computational scheme
can be arranged in a way that each computing node carries out an independent task
(a copy of the Cambridge Quantum Monte Carlo Casino code is available at http://
vallico.net/casinoqmc/).
3.1.3 Many-Electron Wavefunctions
Yet, the most popular approach to the problem of producing many-electron wavefunctions, starts from the corresponding single electron (hydrogen-like or their variants)
solution of the time-independent Schrödinger equation given in Eq. 2.18. In fact,
even if the single electron hydrogen-like wavefunctions computed in this way suffer
severe limitations, they provide us with a conceptual road map that has actually led
us in the past to construct many-electron wavefunctions of increasing accuracy. This
is obtained by progressively integrating the Hamiltonian with the terms necessary
to describe the additional complexity of molecular processes including reactions.
For example, it has to be noted here that the two-body (nucleus + electron) system considered in Eq. 2.18 is inadequate to describe many-electron systems due to
the insufficiency of the conservation of the classical angular momentum alone for
quantum systems. This means that an additional component s (and the related spin
quantum number valued either 1/2 or −1/2) needs to be introduced in the definition
of the electronic wavefunction due to the fermionic nature of electrons. Accordingly,
we shall consider the χ nlms wavefunctions (the spinorbitals) associated with electrons
assigned to “stable orbits of discrete energies” and characterized by the set of four
quantum numbers (n, l, m and s), with no two of them having the same set of four
values, as the basic building blocks of any quantum formulations of the chemical
processes.
This already allows us not only to correctly describe the manifold of the excited
atomic electronic states but also provides us with a proper formulation of the manyelectron (say K -electronic) wavefunctions (r) as a product of the χ nlms spinor-
3 Ab initio Electronic Structure for Few-Body Systems
In the VMC method, therefore, it is crucial the choice of the tentative T (r) that
determines the value of the observable computed by the simulation.
On the contrary, the DMC approach avoids the need for setting an approximate
tentative wavefunction by resorting to the use of the time-dependent formalism
H(r, τ ) = −
∂(r, τ )
∂τ
(3.4)
where τ is the imaginary time (τ = it). In Eq. 3.4 (r, τ ) tends to relax to the true
ground state wavefunction as τ → ∞ even when one begins with any arbitrary function at τ = 0 ((R, τ = 0)). In other words, the DMC trial wavefunction converges
in any case to the numerically accurate solution and its propagation is only a mathematical expedient when the algorithm is accurate and efficient. Accordingly, one
can drop τ from the notation when not strictly necessary. In addition, the method is
of particular interest for distributed computing because the computational scheme
can be arranged in a way that each computing node carries out an independent task
(a copy of the Cambridge Quantum Monte Carlo Casino code is available at http://
vallico.net/casinoqmc/).
3.1.3 Many-Electron Wavefunctions
Yet, the most popular approach to the problem of producing many-electron wavefunctions, starts from the corresponding single electron (hydrogen-like or their variants)
solution of the time-independent Schrödinger equation given in Eq. 2.18. In fact,
even if the single electron hydrogen-like wavefunctions computed in this way suffer
severe limitations, they provide us with a conceptual road map that has actually led
us in the past to construct many-electron wavefunctions of increasing accuracy. This
is obtained by progressively integrating the Hamiltonian with the terms necessary
to describe the additional complexity of molecular processes including reactions.
For example, it has to be noted here that the two-body (nucleus + electron) system considered in Eq. 2.18 is inadequate to describe many-electron systems due to
the insufficiency of the conservation of the classical angular momentum alone for
quantum systems. This means that an additional component s (and the related spin
quantum number valued either 1/2 or −1/2) needs to be introduced in the definition
of the electronic wavefunction due to the fermionic nature of electrons. Accordingly,
we shall consider the χ nlms wavefunctions (the spinorbitals) associated with electrons
assigned to “stable orbits of discrete energies” and characterized by the set of four
quantum numbers (n, l, m and s), with no two of them having the same set of four
values, as the basic building blocks of any quantum formulations of the chemical
processes.
This already allows us not only to correctly describe the manifold of the excited
atomic electronic states but also provides us with a proper formulation of the manyelectron (say K -electronic) wavefunctions (r) as a product of the χ nlms spinor-