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.

Question

Derive eq293.

Answer

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

$E_{est}=-\frac{m_ee^4}{32\pi^2\varepsilon_0^{\;2}\hbar^2}$

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.

Question

Provide a proof of the variational principle.

Answer

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, $E_{\psi}\geq E_0$ .

The variational method

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