en>Yobot: Reference before punctuation and other fixes using AWB (9510)

2013-10-04T14:21:17Z

Reference before punctuation and other fixes using AWB (9510)

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{{Multiple issues|unreferenced = January 2010|unreferenced = December 2009}}

In [[quantum chemistry]], the '''Dyall Hamiltonian''' is a modified [[Hamiltonian (quantum mechanics)|Hamiltonian]] with two-electron nature. It can be written as follows:

:<math>\hat{\mathcal{H}}^D = \hat{\mathcal{H}}^D_i + \hat{\mathcal{H}}^D_v + C</math>
:<math>\hat{\mathcal{H}}^D_i = \sum_{i}^{\rm core} \epsilon_i E_{ii} + \sum_r^{\rm virt} \epsilon_r E_{rr} </math>
:<math>\hat{\mathcal{H}}^D_v = \sum_{ab}^{\rm act} h_{ab}^{\rm eff} E_{ab} +
\frac{1}{2} \sum_{abcd}^{\rm act} \left\langle ab \left.\right| cd \right\rangle \left(E_{ac}
E_{bd} - \delta_{bc} E_{ad} \right)</math>
:<math>C = 2 \sum_{i}^{\rm core} h_{ii} + \sum_{ij}^{\rm core} \left( 2 \left\langle ij \left.\right| ij\right\rangle - \left \langle ij \left.\right| ji\right\rangle \right) - 2 \sum_{i}^{\rm core} \epsilon_i</math>
:<math>h_{ab}^{\rm eff} = h_{ab} + \sum_j \left( 2 \left\langle aj \left.\right| bj \right\rangle -
\left\langle aj \left.\right| jb \right\rangle \right)</math>

where labels <math>i,j,\ldots</math>, <math>a,b,\ldots</math>, <math>r,s,\ldots</math> denote core, active and virtual orbitals (see [[Complete active space]]) respectively, <math>\epsilon_i</math> and <math>\epsilon_r</math> are the orbital energies of the involved orbitals, and <math>E_{mn}</math> operators are the spin-traced operators <math>a^{\dagger}_{m\alpha}a_{n\alpha} + a^{\dagger}_{m\beta}a_{n\beta}</math>. These operators commute with <math>S^2</math> and <math>S_z</math>, therefore the application of these operators on a spin-pure function produces again a spin-pure function.

The Dyall Hamiltonian behaves like the true Hamiltonian inside the CAS space, having the same eigenvalues and eigenvectors of the true Hamiltonian projected onto the CAS space.

[[Category:Quantum chemistry]]

{{theoretical-chem-stub}}

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en>Yobot: Reference before punctuation and other fixes using AWB (9510)