12: Benzene — Valence and low-lying conduction states¶
Valence States¶
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Outline: Obtain MLWFs for the valence bands of benzene.
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Items 1-4: Inspect the output file benzene.wout. The total spread converges to its minimum value after just a few iterations.
Convergence of total spread \(\Omega\) is shown in Figure 2. The spread converges very quickly, and after only 15 iterations the \(|\Delta\Omega|\) is already below \(10^{-8}\).
Convergence of total spread \(\Omega\). The red curve refers to the left y-axis, i.e. the actual value of the total spread at each iteration. The blue curve refers to the right y-axis, i.e. the absolute difference between between the spread functional at iteration \(i\) and \(i-1\), i.e. \(\Delta\Omega\). Below is shown the final state of the minimization, after 22 iterations:
Output fileFinal State WF centre and spread 1 ( -6.875141, 7.935472, 7.937658 ) 0.65233309 WF centre and spread 2 ( 4.748245, 7.935472, 7.937658 ) 0.65234300 WF centre and spread 3 ( 5.809324, 6.097678, 7.937658 ) 0.65186298 WF centre and spread 4 ( 7.816785, -7.396927, 7.648002 ) 1.20135958 WF centre and spread 5 ( 7.816785, -7.396927, -7.648002 ) 1.20135948 WF centre and spread 6 ( 7.936083, 7.324218, -7.937658 ) 0.60992987 WF centre and spread 7 ( 7.935073, -6.095184, 7.937658 ) 0.65161009 WF centre and spread 8 ( 6.874208, -6.712573, 7.937658 ) 0.61134798 WF centre and spread 9 ( 6.874224, 6.850489, -7.648754 ) 1.20500764 WF centre and spread 10 ( -7.936214, 6.097679, 7.937658 ) 0.65186256 WF centre and spread 11 ( 5.813353, -6.095183, 7.937658 ) 0.65160955 WF centre and spread 12 ( 6.874224, 6.850489, 7.648754 ) 1.20500766 WF centre and spread 13 ( 5.812342, 7.324217, 7.937658 ) 0.60993084 WF centre and spread 14 ( 5.931632, -7.396946, 7.648001 ) 1.20136423 WF centre and spread 15 ( 5.931632, -7.396946, -7.648001 ) 1.20136424 Sum of centres and spreads ( 71.362557, 7.925029, 55.563607 ) 12.95829277 Spreads (Ang^2) Omega I = 10.455434168 ================ Omega D = 0.000000000 Omega OD = 2.502858604 Final Spread (Ang^2) Omega Total = 12.958292772 ------------------------------------------------------------------------------ -
Item 5: Plot the MLWFs 2-4.
MLWFs are shown in Figure 3.
MLWFs 2, 3 and 4, with Vesta from cubeformat.
Valence + Conduction States¶
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Outline: Obtain MLWFs for the valence and low-lying conduction states of benzene.
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Item 1: First, we minimise \(\Omega_I\). Then, we minimise \(\Omega_D + \Omega_{OD}\).
Extract from the
.woutoutput file for the disentanglement procedure with initial and final value of \(\Omega_I\)Output fileExtraction of optimally-connected subspace ------------------------------------------ +---------------------------------------------------------------------+<-- DIS | Iter Omega_I(i-1) Omega_I(i) Delta (frac.) Time |<-- DIS +---------------------------------------------------------------------+<-- DIS 1 14.77292507 14.36793746 2.819E-02 0.06 <-- DIS . . . . . . . . . . 76 14.26979011 14.26979011 7.234E-11 0.34 <-- DIS <<< Delta < 1.000E-10 over 3 iterations >>> <<< Disentanglement convergence criteria satisfied >>> Final Omega_I 14.26979011 (Ang^2) +----------------------------------------------------------------------------+Below a snippet from the
.woutoutput file, showing the finale state of the minimisation of \(\Omega_D\) and \(\Omega_{OD}\).Output fileFinal State WF centre and spread 1 ( -6.872991, -7.937658, 7.937657 ) 0.64685210 WF centre and spread 2 ( -7.937084, -6.094542, -7.937656 ) 0.64620663 WF centre and spread 3 ( 5.810194, -6.094542, 7.937654 ) 0.64620672 WF centre and spread 4 ( 4.746097, -7.937658, 7.937657 ) 0.64685949 WF centre and spread 5 ( 5.810193, 6.094541, -7.937657 ) 0.64620720 WF centre and spread 6 ( -7.937083, 6.094541, 7.937655 ) 0.64620743 WF centre and spread 7 ( 6.874209, 6.725491, 7.937658 ) 0.58709128 WF centre and spread 8 ( 6.874209, -6.725491, 7.937657 ) 0.58709059 WF centre and spread 9 ( 5.824168, -7.331141, 7.937657 ) 0.58581741 WF centre and spread 10 ( 5.824168, 7.331141, -7.937658 ) 0.58581778 WF centre and spread 11 ( 7.924257, 7.331142, 7.937657 ) 0.58581692 WF centre and spread 12 ( 7.924257, -7.331141, -7.937658 ) 0.58581674 WF centre and spread 13 ( -7.514755, 7.937658, -7.937656 ) 1.57077288 WF centre and spread 14 ( 7.622957, -6.642249, 7.937655 ) 1.58413409 WF centre and spread 15 ( 6.125499, -6.642276, -7.937651 ) 1.58406298 WF centre and spread 16 ( 5.387848, -7.937658, -7.937656 ) 1.57079972 WF centre and spread 17 ( 6.125498, 6.642275, 7.937656 ) 1.58407213 WF centre and spread 18 ( 7.622957, 6.642249, -7.937653 ) 1.58414266 Sum of centres and spreads ( 60.234596,-15.875317, 15.875317 ) 16.87397474 Spreads (Ang^2) Omega I = 14.269790106 ================ Omega D = 0.000000000 Omega OD = 2.604184635 Final Spread (Ang^2) Omega Total = 16.873974742 ------------------------------------------------------------------------------ -
Item 2: Plot the MLWFs 1, 7 and 13.
MLWFs are shown in Figure 4.
MLWFs 1, 7 and 13 with Vesta from cubeformat.