H equals the triple sum over i, alpha, sigma of epsilon sub i, alpha times c dagger sub i, alpha, sigma times c sub i, alpha, sigma plus the sum i j over nearest neighbors of the triple sum over alpha, beta, sigma of t sub i alpha , j beta times c dagger sub i, alpha, sigma times c sub j, beta, sigma plus the five fold sum over i, alpha, beta, sigma, eta of U sub i alpha, j beta times ( one minus the product of kronecker delta of alpha , beta times kronecker delta of sigma, eta ) times c dagger sub i, alpha, sigma times c sub i, alpha, sigma times c dagger sub i, beta, eta times c sub i, beta, eta

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