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When dealing with certain quantum systems, an absolutely quantitative and accurate description of the system is impossible and requires physicists and chemists to make approximations. For example, one may calculate the Hamiltonian of a single hydrogen atom or a molecule of diatomic helium with a single electron (after invoking the Born-Oppenheimer Approximation of course), but cannot solve a multi-electron problem such as benzene.
Although we cannot calculate the Hamiltonian for benzene, we can approximate it and receive an answer which is very close (and according to the Variation Principal, higher than) the actual energy (Hamiltonian).
One way that computational chemists do this is by using variational approximations. One of these which is most popular is the Hartree-Fock method. Here, chemists say that there exists a ground state wavefunction which describes the benzene system that may be approximated by a single Slater Determinant. We chose a candidate wavefunction which we think suits the system (think e^ikx for SHOs) and which depends on a set of parameters. We then calculate the Hamiltonian for sets of parameters and find the lowest energy. This is a gross oversimplification, but the idea holds.
A simpler way to think about this would be: "What is the shape of a rope tied to a bucket of water?"
We could answer this question by starting with an equation for the rope in 2 dimensions, calculate the potential energy of the bucket as the rope changes coordinates, and eventually find that it's potential energy is minimized when the rope extends completely along the y axis.
Variational approximations work quite the same way for quantum systems where, due to the entangled nature of quantized particles (such as fermions or bosons) we cannot derive an exact answer.
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ReneVariational approximation is a method used to approximate the wavefunction of a quantum system by varying a set of parameters to minimize the energy. It aims to find a wavefunction that is close to the exact solution, but is simpler and easier to solve for. This method is commonly used to estimate properties of quantum systems when an exact solution is not possible.
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