The variational method

The variational method is a mathematical technique for approximating the energy state of a system, most often the ground state of a multi-electron system.

To illustrate the method, we begin with the expectation value for the ground state energy of a system, E_0:

E_0=\frac{\int\psi_0^*\hat{H}\psi_0d\tau}{\int\psi_0^*\psi_0d\tau}\; \; \; \; \; \; \; \; 293

where \psi_0 is the ground state wavefunction of the system.



Derive eq293.


Eq293 is obtained by multiplying the Schrodinger equation from the left by \psi_0^* and integrating over all space.


As mentioned in an earlier article, E_0 is, by definition, the lowest energy value of the system. If we replace \psi_0 (the mathematical description of the ground state energy of the system) with any wave function \psi that describes any energy state of the system,

E_{\psi}=\frac{\int\psi^*\hat{H}\psi d\tau}{\int\psi^*\psi d\tau}\geq E_0\; \; \; \; \; \; \; \; 294

where E_{\psi}, E_0\leftrightarrow \psi,\psi_0.

If \psi is normalised,

E_{\psi}=\int\psi^*\hat{H}\psi d\tau\geq E_0\; \; \; \; \; \; \; \; 295

Eq295 is called the variational principle, i.e. the principle behind the variational method. It is evident that eq295 is a functional: a function E_{\psi} of a function \psi. Therefore, to estimate the ground state energy of a system, we substitute an appropriate trial wave function that depends on one or more arbitrary parameters (variational parameters) into eq295, and minimise E_{\psi} with respect to the parameters.

For example, we can estimate the ground state energy of a hydrogen atom using the trial wavefunction \psi=e^{-\alpha r}, where \alpha is the variational parameter. Substituting \psi=e^{-\alpha r}, \hat{H}=-\frac{\hbar^2}{2m_er^2}\frac{d}{dr}\left ( r^2\frac{d}{dr} \right )-\frac{e^2}{4\pi\varepsilon_0r} in eq295, and using the identity \int_0^{\infty}x^ne^{-a x}dx=\frac{n!}{a^{n+1}}, we have

E_{\psi}=\frac{\hbar^2\alpha^2}{2m_e}-\frac{e^2\alpha}{4\pi\varepsilon_0}\; \; \; \; \; \; \; \; 296

Clearly, eq296 describes a parabola with a minimum value. Setting \frac{dE_{\psi}}{d\alpha}=0, solving for \alpha, and substituting the expression for \alpha back into eq296, we have


where E_{est} is the estimated ground state energy of hydrogen.

In other words, we vary the variational parameter to arrive at a good approximation of E_{\psi}.

To determine the ground state energy of a multi-electron system, the variational method is used together with the Hartree self-consistent method, which will be explained in subsequent articles.



Provide a proof of the variational principle.


Since any well-behaved wavefunction can be expressed as a linear combination of a complete orthonormal set of basis wavefunctions, we substitute \psi=\sum_{n=0}^{\infty}c_n\phi_n in eq295.

E_{\psi}=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\int c_m^*\phi_m^*\hat{H}c_n\phi_nd\tau=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}c_m^*c_nE_n\int\phi_m^*\phi_nd\tau=\sum_{m=0}^{\infty}\vert c_m\vert^2E_m

Since \sum_{m=0}^{\infty}\vert c_m\vert^2=1, we have E_0=\sum_{m=0}^{\infty}\vert c_m\vert^2E_0. So,

E_{\psi}-E_0=\sum_{m=0}^{\infty}\vert c_m\vert^2E_m -\sum_{m=0}^{\infty}\vert c_m\vert^2E_0=\sum_{m=0}^{\infty}\vert c_m\vert^2(E_m-E_0)

E_0 is, by definition, the lowest energy value of the system. So, \sum_{m=0}^{\infty}\vert c_m\vert^2(E_m-E_0) \geq 0 and is zero if and only if E_m=E_0. Therefore, E_{\psi}\geq E_0.



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