lecture8 - HW 6 Bloch states in one-dimension When V(x is...

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HW 6, Bloch states in one-dimension • When V(x) is periodic with periodicity L, we get Bloch states • The summation is over the (one-dimensional) vectors • Then we have to solve the linear equations
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Elements of H-matrix in the plane-wave basis • Diagonal elements given by kinetic-energy term • Off-diagonal elements require Fourier transform of V(x) • Use subroutine four1.f to perform Fourier transform • Matrix elements directly come from the FFT of V(x) • Diagonalization for eigenvalues/eigenvectors from ch.f
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General outline of code… 1. Determine input potential V(x) at discrete x j =j Δ x 2. Compute Fourier transform of V(x) by calling four1.f 3. Populate off-diagonal elements of H with FT of V(x) 4. Begin loop on k-points, from - π /L to π /L 5. Populate diagonal matrix elements 6. Store real elements of h(i,j) in hr(i,j), imaginary in hi(i,j) 7. Compute eigenvalues/eigenvectors by calling ch.f 8. Output result for current k-value, return to 4 until done
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Declaration statements… Complex numbers again! IMPLICIT NONE INTEGER, PARAMETER :: Prec14=SELECTED_REAL_KIND(14) INTEGER :: i,j,n,isign,ik,jmi,ierr,matz INTEGER, PARAMETER :: jmax=64,ikmax=20 COMPLEX(KIND=Prec14), DIMENSION(0:jmax-1) :: v ! potential and Fourier tran COMPLEX(KIND=Prec14), DIMENSION(0:jmax-1,0:jmax-1) :: h ! Hamiltonian ma
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lecture8 - HW 6 Bloch states in one-dimension When V(x is...

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