Introduction to DFT: VASP - Center for Autonomous Materials ...

1

Introduction to DFT: VASP

Corey OsesMonday, July 12, 2021

2

Motivation: From theory to practice

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3

Time-independent many-body Schrödinger equation

Born-Oppenheimer approximation: clamped nuclei

mproton

melectron⇡ 1800

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E = Ee + Encl-ncl

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spin-less MANY-BODY wave-function

H ⇣~r1,~r2, . . . ,~rN , ~R1,

~R2, . . . ,~RM

⌘= E

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zAzB

~rAB

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Te

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Tncl

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Ve-e

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Ve-ncl

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Vncl-ncl

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0 const

4

Curse of dimensionality

He e (~r1,~r2, . . . ,~rN ) = Ee e

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solving for single, many-body wave-function (3N coordinates) is expensive

• computational cost grows exponentially with dimensionality:

example: 3D sampling grid with 10 points along each direction

cost factor relative to that of an electron in 1D:

• CO2 molecule:

• nanocluster of 100 platinum atoms:

Bellman, Princeton Univ. Press (1957)

(ngrid)3N

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�103

�3⇥22= 10198

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�103

�3⇥78⇥100= 1070,200

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E =

Z. . .

Z ⇤

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3

5 d~r1d~r2 . . . d~rN

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Density-functional theory

5

E [ (~r1,~r2, . . . ,~rN )] ! E [⇢ (~r)]

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Hohenberg-Kohn (1964) establisheswithout defining the exact form of the equations

accounts for: residual T of interacting particles, exchange energy, correlation energy, self-interaction

PRB 23, 5048 (1981); Parr and Yang, Oxford Univ. Press (1994)

describes N non-interacting particles

classic Coulombenergy

garbage collection

= Ts [⇢ (~r)] + UH [⇢ (~r)] +

ZVext (~r) ⇢ (~r) d~r + Exc [⇢ (~r)]

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Exc (⇢ (~r)) = (T [⇢ (~r)]� Ts [⇢ (~r)]) + (Ue-e [⇢ (~r)]� UH [⇢ (~r)])

<latexit sha1_base64="jaMh+D6Wb8xZ4rd4nz8LCXCktiM=">AAADB3ichVJNSxxBEO0ZjR9jYlZzDEKTJWAILjNhQT0EhCB4VNhVYWZYenprdht7PuiuEZdhbrnkr+SSQ0Lwmr/gLf8mvbszYjTRgqYfr+rV667uKJdCo+v+tuyFxWdLyyurztrzF+svWxubpzorFIc+z2SmziOmQYoU+ihQwnmugCWRhLPo4tM0f3YJSoss7eEkhzBho1TEgjM01GDD2joclAHCFZZXvKoCCTFuB2qc1egSeKmqQInRGN/V20dnnqRObwb8x+tD6uzQXuOiaxOf/k9FnVtdzTjv53VOv+kCO9D0ecLceN+qjp70bkSN8aDVdjvuLOhD4NWgTeo4HrRugmHGiwRS5JJp7XtujmHJFAouoXKCQkPO+AUbgW9gyhLQYTl7x4q+NcyQxpkyK0U6Y+8qSpZoPUkiU5kwHOv7uSn5r5xfYLwXliLNC4SUz43iQlLM6PRT0KFQwFFODGBcCXNWysdMMY7m60yH4N2/8kNw+qHjdTv7J932Qbcexwp5Td6QbeKRXXJAjsgx6RNufba+Wt+tH/YX+5v9076el9pWrXlF/gr71x+l1Piq</latexit>

= T [⇢ (~r)] + Ue-e [⇢ (~r)] +

ZVext (~r) ⇢ (~r) d~r

<latexit sha1_base64="I2NmmleD/VZU8M4tgqOk4ax5454=">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</latexit>

implied by H-K

Kohn-Sham (1965) offers a useful reformulation:

�1

2r2 + ve↵

� i (~r) = ✏i i (~r)

<latexit sha1_base64="XVZgmrUfdlrxb0N6KlaO72wpRVw=">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</latexit>

ve↵ =

Z⇢(~rj)

|~ri � ~rj |d~rj + Vext + Vxc

<latexit sha1_base64="flQQ1q/BM+GhMPSAlF7lcELlEOs=">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</latexit>

⇢ (~r) =NX

i

| i (~r) |2

<latexit sha1_base64="hShGAiGyvpiTYamAcgSODUrYEwk=">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</latexit>

NON-LINEAR eigenvalue problem: solve iteratively (see OSZICAR)

1. set

2. plug into

3. solve for (normal eigenvalue problem)

4. while

1. update

2. plug into

3. solve for (normal eigenvalue problem)

<latexit sha1_base64="dqMa/bBDECM1oX0PH5ly9b2QmsA=">AAAB63icbVBNSwMxEJ3Ur1q/qh69BIvgqexKQb0VvHisYD+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5YSK4sZ73jUobm1vbO+Xdyt7+weFR9fikY+JUU9amsYh1LySGCa5Y23IrWC/RjMhQsG44vcv97hPThsfq0c4SFkgyVjzilNhcGiSGD6s1r+4tgNeJX5AaFGgNq1+DUUxTyZSlghjT973EBhnRllPB5pVBalhC6JSMWd9RRSQzQba4dY4vnDLCUaxdKYsX6u+JjEhjZjJ0nZLYiVn1cvE/r5/a6CbIuEpSyxRdLopSgW2M88fxiGtGrZg5Qqjm7lZMJ0QTal08FReCv/ryOulc1f1G/fahUWs2ijjKcAbncAk+XEMT7qEFbaAwgWd4hTck0Qt6Rx/L1hIqZk7hD9DnDyPpjkg=</latexit>

0 = trial

<latexit sha1_base64="Ko6TYIx5DRQwMFdQZ4d4lOfbeWQ=">AAACDXicbVDLSsNAFJ3UV62vqks3g1VwVRIpqAuh4MZlBfuApobJ9KYdOnkwcyOWkB9w46+4caGIW/fu/BuTtgttPXDhzDn3MvceN5JCo2l+G4Wl5ZXVteJ6aWNza3unvLvX0mGsODR5KEPVcZkGKQJookAJnUgB810JbXd0lfvte1BahMEtjiPo+WwQCE9whpnklI/sSIu7xI6U8CGllzR/O4mN8IAJKsFkmpaccsWsmhPQRWLNSIXM0HDKX3Y/5LEPAXLJtO5aZoS9hCkUXEJasmMNEeMjNoBuRgPmg+4lk2tSepwpfeqFKqsA6UT9PZEwX+ux72adPsOhnvdy8T+vG6N33ktEEMUIAZ9+5MWSYkjzaGhfKOAoxxlhXIlsV8qHTDGOWYB5CNb8yYukdVq1atWLm1qlXpvFUSQH5JCcEIuckTq5Jg3SJJw8kmfySt6MJ+PFeDc+pq0FYzazT/7A+PwBOA6cQw==</latexit>

0 =

<latexit sha1_base64="mQDh2o9RyaM6vUaP6lMsClCMpAk=">AAAB/XicbVDLSgMxFM34rPU1PnZugkVwVWakoC6EghuXFewDOmPJpJk2NMmEJCPUofgrblwo4tb/cOffmGlnoa0HLpyccy+590SSUW0879tZWl5ZXVsvbZQ3t7Z3dt29/ZZOUoVJEycsUZ0IacKoIE1DDSMdqQjiESPtaHSd++0HojRNxJ0ZSxJyNBA0phgZK/Xcw0Bqep8FUlFOJvAK5u+eW/Gq3hRwkfgFqYACjZ77FfQTnHIiDGZI667vSRNmSBmKGZmUg1QTifAIDUjXUoE40WE23X4CT6zSh3GibAkDp+rviQxxrcc8sp0cmaGe93LxP6+bmvgizKiQqSECzz6KUwZNAvMoYJ8qgg0bW4KwonZXiIdIIWxsYGUbgj9/8iJpnVX9WvXytlap14o4SuAIHINT4INzUAc3oAGaAINH8AxewZvz5Lw4787HrHXJKWYOwB84nz87NZUN</latexit>

0

<latexit sha1_base64="1B50pr8lERtZo5cg09z5HlDv2G8=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9mVgnorePFYwbZCu5Zsmm1Dk2xMsoWy9Hd48aCIV3+MN/+NabsHbX0w8Hhvhpl5keLMWN//9gpr6xubW8Xt0s7u3v5B+fCoZZJUE9okCU/0Q4QN5UzSpmWW0welKRYRp+1odDPz22OqDUvkvZ0oGgo8kCxmBFsnhV1l2GPWVZoJOu2VK37VnwOtkiAnFcjR6JW/uv2EpIJKSzg2phP4yoYZ1pYRTqelbmqowmSEB7TjqMSCmjCbHz1FZ07pozjRrqRFc/X3RIaFMRMRuU6B7dAsezPxP6+T2vgqzJhUqaWSLBbFKUc2QbMEUJ9pSiyfOIKJZu5WRIZYY2JdTiUXQrD88ippXVSDWvX6rlap1/I4inACp3AOAVxCHW6hAU0g8ATP8Apv3th78d69j0VrwctnjuEPvM8fVF+ScQ==</latexit>

<latexit sha1_base64="dqMa/bBDECM1oX0PH5ly9b2QmsA=">AAAB63icbVBNSwMxEJ3Ur1q/qh69BIvgqexKQb0VvHisYD+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5YSK4sZ73jUobm1vbO+Xdyt7+weFR9fikY+JUU9amsYh1LySGCa5Y23IrWC/RjMhQsG44vcv97hPThsfq0c4SFkgyVjzilNhcGiSGD6s1r+4tgNeJX5AaFGgNq1+DUUxTyZSlghjT973EBhnRllPB5pVBalhC6JSMWd9RRSQzQba4dY4vnDLCUaxdKYsX6u+JjEhjZjJ0nZLYiVn1cvE/r5/a6CbIuEpSyxRdLopSgW2M88fxiGtGrZg5Qqjm7lZMJ0QTal08FReCv/ryOulc1f1G/fahUWs2ijjKcAbncAk+XEMT7qEFbaAwgWd4hTck0Qt6Rx/L1hIqZk7hD9DnDyPpjkg=</latexit>

<latexit sha1_base64="n9qveGY1cTsP3Bw/guVGGFDkNoQ=">AAAB/nicbVDLSgNBEJyNrxhfUfHkZTAInsKuBvUY8OIxgomB7BJmJ73JkNkHM73BsCz4K148KOLV7/Dm3zh5HDSxoKGo6qa7y0+k0Gjb31ZhZXVtfaO4Wdra3tndK+8ftHScKg5NHstYtX2mQYoImihQQjtRwEJfwoM/vJn4DyNQWsTRPY4T8ELWj0QgOEMjdctH7oBhNsq7mYvwiBkEQZ7TbrliV+0p6DJx5qRC5mh0y19uL+ZpCBFyybTuOHaCXsYUCi4hL7mphoTxIetDx9CIhaC9bHp+Tk+N0qNBrExFSKfq74mMhVqPQ990hgwHetGbiP95nRSDay8TUZIiRHy2KEglxZhOsqA9oYCjHBvCuBLmVsoHTDGOJrGSCcFZfHmZtM6rzmX14q5WqdfmcRTJMTkhZ8QhV6RObkmDNAknGXkmr+TNerJerHfrY9ZasOYzh+QPrM8fNB+WPw==</latexit>

ve↵

<latexit sha1_base64="mRijGH9Y+/akxU/ygXHEGGkec90=">AAAB/3icbZDLSsNAFIYn9VbrLSq4cTNYBFcl0aIuC25cVrAXaGKZTE/aoZNJnJkIJXbhq7hxoYhbX8Odb+Ok7UJbfxj4+M85nDN/kHCmtON8W4Wl5ZXVteJ6aWNza3vH3t1rqjiVFBo05rFsB0QBZwIammkO7UQCiQIOrWB4lddbDyAVi8WtHiXgR6QvWMgo0cbq2gdeopgn4B7ncJd5iWQRjLt22ak4E+FFcGdQRjPVu/aX14tpGoHQlBOlOq6TaD8jUjPKYVzyUgUJoUPSh45BQSJQfja5f4yPjdPDYSzNExpP3N8TGYmUGkWB6YyIHqj5Wm7+V+ukOrz0MyaSVIOg00VhyrGOcR4G7jEJVPORAUIlM7diOiCSUG0iK5kQ3PkvL0LztOKeV85uquVadRZHER2iI3SCXHSBauga1VEDUfSIntErerOerBfr3fqYthas2cw++iPr8wd9wpZg</latexit>

6= 0

0

<latexit sha1_base64="1B50pr8lERtZo5cg09z5HlDv2G8=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9mVgnorePFYwbZCu5Zsmm1Dk2xMsoWy9Hd48aCIV3+MN/+NabsHbX0w8Hhvhpl5keLMWN//9gpr6xubW8Xt0s7u3v5B+fCoZZJUE9okCU/0Q4QN5UzSpmWW0welKRYRp+1odDPz22OqDUvkvZ0oGgo8kCxmBFsnhV1l2GPWVZoJOu2VK37VnwOtkiAnFcjR6JW/uv2EpIJKSzg2phP4yoYZ1pYRTqelbmqowmSEB7TjqMSCmjCbHz1FZ07pozjRrqRFc/X3RIaFMRMRuU6B7dAsezPxP6+T2vgqzJhUqaWSLBbFKUc2QbMEUJ9pSiyfOIKJZu5WRIZYY2JdTiUXQrD88ippXVSDWvX6rlap1/I4inACp3AOAVxCHW6hAU0g8ATP8Apv3th78d69j0VrwctnjuEPvM8fVF+ScQ==</latexit>

<latexit sha1_base64="n9qveGY1cTsP3Bw/guVGGFDkNoQ=">AAAB/nicbVDLSgNBEJyNrxhfUfHkZTAInsKuBvUY8OIxgomB7BJmJ73JkNkHM73BsCz4K148KOLV7/Dm3zh5HDSxoKGo6qa7y0+k0Gjb31ZhZXVtfaO4Wdra3tndK+8ftHScKg5NHstYtX2mQYoImihQQjtRwEJfwoM/vJn4DyNQWsTRPY4T8ELWj0QgOEMjdctH7oBhNsq7mYvwiBkEQZ7TbrliV+0p6DJx5qRC5mh0y19uL+ZpCBFyybTuOHaCXsYUCi4hL7mphoTxIetDx9CIhaC9bHp+Tk+N0qNBrExFSKfq74mMhVqPQ990hgwHetGbiP95nRSDay8TUZIiRHy2KEglxZhOsqA9oYCjHBvCuBLmVsoHTDGOJrGSCcFZfHmZtM6rzmX14q5WqdfmcRTJMTkhZ8QhV6RObkmDNAknGXkmr+TNerJerHfrY9ZasOYzh+QPrM8fNB+WPw==</latexit>

ve↵

Density-functional theory

6

canonical Kohn-Sham orbital equations: N one-electron equations

Density-functional theory in crystals

7

DFT cut costs down significantly: (ngrid)3N ! N ⇥ (ngrid)

3

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reduces much further in crystals

“macroscopic” material

nanoscopic crystal

what about materials? N ⇠ O(1023)

<latexit sha1_base64="+AXFWvI7YTcN0yDpVFeksu5BQRs=">AAACBHicdVDLSsNAFJ34rPUVddnNYBHqpiRp6GNXcONKK9gHNLVMptN26GQSZiZCCV248VfcuFDErR/hzr9x0lZQ0QMXDufcy733+BGjUlnWh7Gyura+sZnZym7v7O7tmweHLRnGApMmDlkoOj6ShFFOmooqRjqRICjwGWn7k7PUb98SIWnIr9U0Ir0AjTgdUoyUlvpm7gJ6kgbQC5AaY8SSy1nBtm4SpzQ77Zt5q1irlh23DK2iZVVsx06JU3FLLrS1kiIPlmj0zXdvEOI4IFxhhqTs2lakegkSimJGZlkvliRCeIJGpKspRwGRvWT+xAyeaGUAh6HQxRWcq98nEhRIOQ183ZneKn97qfiX143VsNpLKI9iRTheLBrGDKoQponAARUEKzbVBGFB9a0Qj5FAWOncsjqEr0/h/6TlFG23WLty83V3GUcG5MAxKAAbVEAdnIMGaAIM7sADeALPxr3xaLwYr4vWFWM5cwR+wHj7BBXalxY=</latexit>

~R = n1~a1 + n2~a2 + n3~a3

<latexit sha1_base64="+ojylNnWQBSeXGje0UMS29pzh5s=">AAACI3icbZDLSsNAFIYn9VbrLerSTbAIglCStuAFhIIbl1XsBdoQJtNJO3QyCTOTQgl5Fze+ihsXSnHjwndxkgaqrT8M/POdc5g5vxtSIqRpfmmFtfWNza3idmlnd2//QD88aosg4gi3UEAD3nWhwJQw3JJEUtwNOYa+S3HHHd+l9c4Ec0EC9iSnIbZ9OGTEIwhKhRz9pj/BKH5MbpkTW0l2gUlqLxSoLkA1A7UFqCWOXjYrZiZj1Vi5KYNcTUef9QcBinzMJKJQiJ5lhtKOIZcEUZyU+pHAIURjOMQ9ZRn0sbDjbMfEOFNkYHgBV4dJI6O/J2LoCzH1XdXpQzkSy7UU/lfrRdK7smPCwkhihuYPeRE1ZGCkgRkDwjGSdKoMRJyovxpoBDlEUsVaUiFYyyuvmna1YtUr1w/1cqOex1EEJ+AUnAMLXIIGuAdN0AIIPINX8A4+tBftTZtpn/PWgpbPHIM/0r5/AA9rpb0=</latexit>

translational periodicity:

Reciprocal space and lattice

8

\

periodicity modeled with plane waves

where Bloch vector lives in “frequency” domain: reciprocal space

expi~k ~R

<latexit sha1_base64="Di6dxrcgdTjzzlNWEXyQYcuNryc=">AAAB/3icbVDLSsNAFJ34rPUVFdy4CRbBVUmkoO4KblxWsQ9oYplMb9qhk0mYmRRLzMJfceNCEbf+hjv/xmmahbYeuNzDOfcyd44fMyqVbX8bS8srq2vrpY3y5tb2zq65t9+SUSIINEnEItHxsQRGOTQVVQw6sQAc+gza/uhq6rfHICSN+J2axOCFeMBpQAlWWuqZhy48xPcpdcdA0lGWt9ss65kVu2rnsBaJU5AKKtDomV9uPyJJCFwRhqXsOnasvBQLRQmDrOwmEmJMRngAXU05DkF6aX5/Zp1opW8FkdDFlZWrvzdSHEo5CX09GWI1lPPeVPzP6yYquPBSyuNEASezh4KEWSqypmFYfSqAKDbRBBNB9a0WGWKBidKRlXUIzvyXF0nrrOrUqpc3tUq9VsRRQkfoGJ0iB52jOrpGDdREBD2iZ/SK3own48V4Nz5mo0tGsXOA/sD4/AEkw5bO</latexit>

~k

<latexit sha1_base64="iK/gUkmoBpGmlGcOYLIReSR67Hc=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KokU1FvBi8cK9gPaUDbbSbtkswm7m0IJ/RFePCji1d/jzX/jts1BWx8MPN6bYWZekAqujet+OxubW9s7u6W98v7B4dFx5eS0rZNMMWyxRCSqG1CNgktsGW4EdlOFNA4EdoLofu53Jqg0T+STmabox3QkecgZNVbq9CfI8mg2qFTdmrsAWSdeQapQoDmofPWHCctilIYJqnXPc1Pj51QZzgTOyv1MY0pZREfYs1TSGLWfL86dkUurDEmYKFvSkIX6eyKnsdbTOLCdMTVjverNxf+8XmbCWz/nMs0MSrZcFGaCmITMfydDrpAZMbWEMsXtrYSNqaLM2ITKNgRv9eV10r6uefXa3WO92qgXcZTgHC7gCjy4gQY8QBNawCCCZ3iFNyd1Xpx352PZuuEUM2fwB87nD53Mj7s=</latexit>

⌦ = ~a1 · ~a2 ⇥ ~a3

<latexit sha1_base64="gSzCoLjITggALGI6KPFLzQEDm74=">AAACHXicbVDJSgNBEO1xjXGLevTSGARPYSYG1IMQ8OLNCGaBTAg9nUrSpGehuyYQhvkRL/6KFw+KePAi/o2dhaCJDxpevVdFdT0vkkKjbX9bK6tr6xubma3s9s7u3n7u4LCmw1hxqPJQhqrhMQ1SBFBFgRIakQLmexLq3uBm7NeHoLQIgwccRdDyWS8QXcEZGqmdK7l3PvTYtTsEnrC0nTipyzshzuti6qLwQdO5cp62c3m7YE9Al4kzI3kyQ6Wd+3Q7IY99CJBLpnXTsSNsJUyh4BLSrBtriBgfsB40DQ2YWdhKJtel9NQoHdoNlXkB0on6eyJhvtYj3zOdPsO+XvTG4n9eM8buZSsRQRQjBHy6qBtLiiEdR0U7QgFHOTKEcSXMXynvM8U4mkCzJgRn8eRlUisWnFLh6r6UL5dmcWTIMTkhZ8QhF6RMbkmFVAknj+SZvJI368l6sd6tj2nrijWbOSJ/YH39AFS+o0s=</latexit>

real space reciprocal space (1st Brillouin zone)

⌦BZ = ~b1 ·~b2 ⇥ b3

<latexit sha1_base64="hsIcEEs48+xJTB2a5Ouya1bH6As=">AAACJHicbVDJSgNBEO2JW4xb1KOXwSB4CjMxoCJC0Is3I5gFM2Ho6VSSxp6F7hoxDPMxXvwVLx5c8ODFb7GzIJr4oOHVe1VU1/MiwRVa1qeRmZtfWFzKLudWVtfWN/KbW3UVxpJBjYUilE2PKhA8gBpyFNCMJFDfE9Dwbs+HfuMOpOJhcI2DCNo+7QW8yxlFLbn5E+fShx51EwfhHpOzmzQ9de6AJV7qJnbqsE6IP3UpdZD7oEbFQermC1bRGsGcJfaEFMgEVTf/5nRCFvsQIBNUqZZtRdhOqETOBKQ5J1YQUXZLe9DSNKB6VzsZHZmae1rpmN1Q6hegOVJ/TyTUV2rge7rTp9hX095Q/M9rxdg9aic8iGKEgI0XdWNhYmgOEzM7XAJDMdCEMsn1X03Wp5Iy1LnmdAj29MmzpF4q2uXi8VW5UClP4siSHbJL9olNDkmFXJAqqRFGHsgTeSGvxqPxbLwbH+PWjDGZ2SZ/YHx9Aw7ypuY=</latexit>

~G = m1~b1 +m2

~b2 +m3~b3

<latexit sha1_base64="VhaPXWjWoNeB9VL49C+tZ6QJ/aA=">AAACI3icbZDLSsNAFIYn9VbrLerSTbAIglCStuAFhIILXVawF2hDmEwn7dCZJMxMCiXkXdz4Km5cKMWNC9/FSRqotv4w8M93zmHm/G5IiZCm+aUV1tY3NreK26Wd3b39A/3wqC2CiCPcQgENeNeFAlPi45YkkuJuyDFkLsUdd3yX1jsTzAUJ/Cc5DbHN4NAnHkFQKuToN/0JRvF9csuc2Eqyi5uk9kKB6gJUM1BbgFri6GWzYmYyVo2VmzLI1XT0WX8QoIhhXyIKhehZZijtGHJJEMVJqR8JHEI0hkPcU9aHDAs7znZMjDNFBoYXcHV8aWT090QMmRBT5qpOBuVILNdS+F+tF0nvyo6JH0YS+2j+kBdRQwZGGpgxIBwjSafKQMSJ+quBRpBDJFWsJRWCtbzyqmlXK1a9cv1YLzfqeRxFcAJOwTmwwCVogAfQBC2AwDN4Be/gQ3vR3rSZ9jlvLWj5zDH4I+37B/xWpbI=</latexit>

~b1 =2⇡

⌦~a2 ⇥ ~a3

<latexit sha1_base64="oh7CTGPM6q61Ky7h09QPqu4Tjyg=">AAACJHicbVDLSsNAFJ34rPUVdekmWARXJakFFREKbtxZwT6gCWEyvWmHTh7MTAol5GPc+CtuXPjAhRu/xWkaRFsPXDiccy/33uPFjAppmp/a0vLK6tp6aaO8ubW9s6vv7bdFlHACLRKxiHc9LIDREFqSSgbdmAMOPAYdb3Q99Ttj4IJG4b2cxOAEeBBSnxIsleTql/YYSOplbmplV7bPMUlrdkyz1L4NYICz3MbKrmW2pAGIH+E0c/WKWTVzGIvEKkgFFWi6+pvdj0gSQCgJw0L0LDOWToq5pIRBVrYTATEmIzyAnqIhVvucNH8yM46V0jf8iKsKpZGrvydSHAgxCTzVGWA5FPPeVPzP6yXSP3dSGsaJhJDMFvkJM2RkTBMz+pQDkWyiCCacqlsNMsQqKalyLasQrPmXF0m7VrXq1Yu7eqVRL+IooUN0hE6Qhc5QA92gJmohgh7QE3pBr9qj9qy9ax+z1iWtmDlAf6B9fQPIwaa7</latexit>

~b2 =2⇡

⌦~a3 ⇥ ~a1

<latexit sha1_base64="bj3YWyivXaO5Doa8B4/rZRqRnFg=">AAACJHicbVDLSsNAFJ34rPUVdekmWARXJakFFREKbtxZwT6gCWEyvWmHTh7MTAol5GPc+CtuXPjAhRu/xWkaRFsPXDiccy/33uPFjAppmp/a0vLK6tp6aaO8ubW9s6vv7bdFlHACLRKxiHc9LIDREFqSSgbdmAMOPAYdb3Q99Ttj4IJG4b2cxOAEeBBSnxIsleTql/YYSOplblrLrmyfY5LW7JhmqX0bwABnuY2VfZrZkgYgfgQrc/WKWTVzGIvEKkgFFWi6+pvdj0gSQCgJw0L0LDOWToq5pIRBVrYTATEmIzyAnqIhVvucNH8yM46V0jf8iKsKpZGrvydSHAgxCTzVGWA5FPPeVPzP6yXSP3dSGsaJhJDMFvkJM2RkTBMz+pQDkWyiCCacqlsNMsQqKalyLasQrPmXF0m7VrXq1Yu7eqVRL+IooUN0hE6Qhc5QA92gJmohgh7QE3pBr9qj9qy9ax+z1iWtmDlAf6B9fQPJAqa7</latexit>

~b3 =2⇡

⌦~a1 ⇥ ~a2

<latexit sha1_base64="4m5E+qGrvQRWRh5CXZswE1HuhZg=">AAACJHicbVDLSsNAFJ34rPUVdekmWARXJakFFREKbtxZwT6gCWEyvWmHTh7MTAol5GPc+CtuXPjAhRu/xWkaRFsPXDiccy/33uPFjAppmp/a0vLK6tp6aaO8ubW9s6vv7bdFlHACLRKxiHc9LIDREFqSSgbdmAMOPAYdb3Q99Ttj4IJG4b2cxOAEeBBSnxIsleTql/YYSOplbnqaXdk+xySt2THNUvs2gAHOchsr28psSQMQP0Itc/WKWTVzGIvEKkgFFWi6+pvdj0gSQCgJw0L0LDOWToq5pIRBVrYTATEmIzyAnqIhVvucNH8yM46V0jf8iKsKpZGrvydSHAgxCTzVGWA5FPPeVPzP6yXSP3dSGsaJhJDMFvkJM2RkTBMz+pQDkWyiCCacqlsNMsQqKalyLasQrPmXF0m7VrXq1Yu7eqVRL+IooUN0hE6Qhc5QA92gJmohgh7QE3pBr9qj9qy9ax+z1iWtmDlAf6B9fQPJEKa7</latexit>

expi~G·~R = 1

<latexit sha1_base64="kerH+uVVsCc6AGDY+c0dU62iJoA=">AAACBnicbZDLSsNAFIYnXmu9RV2KECyCq5JIQV0IBRe6rGIv0MQymZy0QycXZibFErJy46u4caGIW5/BnW/jNM1CW38Y+PjPOZw5vxszKqRpfmsLi0vLK6ultfL6xubWtr6z2xJRwgk0ScQi3nGxAEZDaEoqGXRiDjhwGbTd4eWk3h4BFzQK7+Q4BifA/ZD6lGCprJ5+YMNDfJ9SewQkvcps4kUy59ssu7B6esWsmrmMebAKqKBCjZ7+ZXsRSQIIJWFYiK5lxtJJMZeUMMjKdiIgxmSI+9BVGOIAhJPmZ2TGkXI8w4+4eqE0cvf3RIoDIcaBqzoDLAditjYx/6t1E+mfOSkN40RCSKaL/IQZMjImmRge5UAkGyvAhFP1V4MMMMdEquTKKgRr9uR5aJ1UrVr1/KZWqdeKOEpoHx2iY2ShU1RH16iBmoigR/SMXtGb9qS9aO/ax7R1QStm9tAfaZ8/8MeZZA==</latexit>

~ai ·~bj = 2⇡�ij

<latexit sha1_base64="vfsJJ7a7L57xdO3E2JYVjMFOzsI=">AAACFHicbVDLSgNBEJyNrxhfUY9eFoMgCGE3BNSDEPDiMYJ5QDaE2dlOMsnsg5neQFj2I7z4K148KOLVgzf/xskmB00sGKiu6qany40EV2hZ30ZubX1jcyu/XdjZ3ds/KB4eNVUYSwYNFopQtl2qQPAAGshRQDuSQH1XQMsd38781gSk4mHwgNMIuj4dBLzPGUUt9YoXzgRYQtNewlOHeSFmtavrUXpTcSLueCCQanuU9oolq2xlMFeJvSAlskC9V/xyvJDFPgTIBFWqY1sRdhMqkTMBacGJFUSUjekAOpoG1AfVTbKjUvNMK57ZD6V+AZqZ+nsiob5SU9/VnT7FoVr2ZuJ/XifG/lU34UEUIwRsvqgfCxNDc5aQ6XEJDMVUE8ok13812ZBKylDnWNAh2Msnr5JmpWxXy9f31VKtuogjT07IKTknNrkkNXJH6qRBGHkkz+SVvBlPxovxbnzMW3PGYuaY/IHx+QPvTJ/5</latexit>

Bloch’s Theorem

9

n

⇣~r,~k

⌘= un

⇣~r,~k

⌘expi

~k~r

<latexit sha1_base64="3g+RtgPkhYhf6f5F1KOEm1WHmF0=">AAACRnichVDLSsNAFL2p7/qqunQTLIKClEQK6kIQ3LhUsCo0sUymN+3QySTM3Igl5OvcuHbnJ7hxoYhb04fgCzwwzOGcc7kzJ0ikMOQ4j1ZpYnJqemZ2rjy/sLi0XFlZvTBxqjk2eCxjfRUwg1IobJAgiVeJRhYFEi+D3vHAv7xBbUSszqmfoB+xjhKh4IwKqVXxvcSIVqZyT2JIW94N8kznO8O7l3tadLq0fZj+l/DwNrnOxKc4yuTlVqXq1Jwh7N/EHZMqjHHaqjx47ZinESrikhnTdJ2E/IxpElxiXvZSgwnjPdbBZkEVi9D42bCG3N4slLYdxro4iuyh+nUiY5Ex/SgokhGjrvnpDcS/vGZK4b6fCZWkhIqPFoWptCm2B53abaGRk+wXhHEtirfavMs041Q0PyjB/fnl3+Rit+bWawdn9epRfVzHLKzDBmyBC3twBCdwCg3gcAdP8AKv1r31bL1Z76NoyRrPrME3lOADYAC0oQ==</latexit>

un

⇣~r + ~R,~k

⌘= un

⇣~r,~k

⌘

<latexit sha1_base64="vy8ysCr21ymNw5tGX1WddY1w5PY=">AAACNHicbVBNS8NAEN3Ur1q/qh69BItQUUoiBfUgCF4ELypWhaaUzXbSLt1swu6kUEJ+lBd/iBcRPCji1d/gNvagrQPLPN57w+w8PxZco+O8WIWZ2bn5heJiaWl5ZXWtvL5xq6NEMWiwSETq3qcaBJfQQI4C7mMFNPQF3Pn9s5F+NwCleSRvcBhDK6RdyQPOKBqqXb5I2qnMPAEBVr0BsFRle3m/zvbz3s88xbs93D2Zdk442uWKU3PysqeBOwYVMq7LdvnJ60QsCUEiE1TrpuvE2EqpQs4EZCUv0RBT1qddaBooaQi6leZHZ/aOYTp2ECnzJNo5+3sipaHWw9A3zpBiT09qI/I/rZlgcNRKuYwTBMl+FgWJsDGyRwnaHa6AoRgaQJni5q8261FFGZqcSyYEd/LkaXB7UHPrteOreuW0Po6jSLbINqkSlxySU3JOLkmDMPJAnskbebcerVfrw/r8sRas8cwm+VPW1zfXlq1V</latexit>

lattice-periodic

n

⇣~r + ~R,~k

⌘= n

⇣~r,~k

⌘expi

~k ~R

<latexit sha1_base64="p0vYA1iZ71g5nlbfTzllDDIlefU=">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</latexit>

constrained to 1st BZ~k

<latexit sha1_base64="iK/gUkmoBpGmlGcOYLIReSR67Hc=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KokU1FvBi8cK9gPaUDbbSbtkswm7m0IJ/RFePCji1d/jzX/jts1BWx8MPN6bYWZekAqujet+OxubW9s7u6W98v7B4dFx5eS0rZNMMWyxRCSqG1CNgktsGW4EdlOFNA4EdoLofu53Jqg0T+STmabox3QkecgZNVbq9CfI8mg2qFTdmrsAWSdeQapQoDmofPWHCctilIYJqnXPc1Pj51QZzgTOyv1MY0pZREfYs1TSGLWfL86dkUurDEmYKFvSkIX6eyKnsdbTOLCdMTVjverNxf+8XmbCWz/nMs0MSrZcFGaCmITMfydDrpAZMbWEMsXtrYSNqaLM2ITKNgRv9eV10r6uefXa3WO92qgXcZTgHC7gCjy4gQY8QBNawCCCZ3iFNyd1Xpx352PZuuEUM2fwB87nD53Mj7s=</latexit>

is band index:n

<latexit sha1_base64="yeUPZxBaysnv+JwfVy5nqJxKZDs=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoN4KXjy2YD+gDWWznbRrN5uwuxFK6C/w4kERr/4kb/4bt20O2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2Akmd3O/84RK81g+mGmCfkRHkoecUWOlphyUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasIbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6tepts1ap1/I4inAG53AJHlxDHe6hAS1ggPAMr/DmPDovzrvzsWwtOPnMKfyB8/kD1ruM8A==</latexit>

O(Ncell)

<latexit sha1_base64="y8Z1flng881ilTEF2HmvRUorbH4=">AAACBnicbVDLSgNBEJyNrxhfUY8iDAYhXsKuBNRbwIsnjWAekIQwO+kkQ2YfzPSKYdmTF3/FiwdFvPoN3vwbZ5McNLGgoajqprvLDaXQaNvfVmZpeWV1Lbue29jc2t7J7+7VdRApDjUeyEA1XaZBCh9qKFBCM1TAPFdCwx1dpn7jHpQWgX+H4xA6Hhv4oi84QyN184dtj+GQMxnfJMXrbtxGeMCYg5RJckK7+YJdsiegi8SZkQKZodrNf7V7AY888JFLpnXLsUPsxEyh4BKSXDvSEDI+YgNoGeozD3QnnryR0GOj9Gg/UKZ8pBP190TMPK3Hnms606P1vJeK/3mtCPvnnVj4YYTg8+mifiQpBjTNhPaEAo5ybAjjSphbKR8yxTia5HImBGf+5UVSPy055dLFbblQKc/iyJIDckSKxCFnpEKuSJXUCCeP5Jm8kjfryXqx3q2PaWvGms3skz+wPn8AdvqZFw==</latexit>

(n > Ncell)

<latexit sha1_base64="KzC2Nat3An7O03TKk7rSKmeKz5A=">AAACB3icbVDLSgNBEJyNrxhfUY+CLAYhXsKuBNSLBLx4kgjmAdkQZie9yZDZ2WWmVwxLbl78FS8eFPHqL3jzb5w8DppY0FBUddPd5ceCa3ScbyuztLyyupZdz21sbm3v5Hf36jpKFIMai0Skmj7VILiEGnIU0IwV0NAX0PAHV2O/cQ9K80je4TCGdkh7kgecUTRSJ3/oCQiwKC9vOqmH8IApAyFGI0/xXh9POvmCU3ImsBeJOyMFMkO1k//yuhFLQpDIBNW65ToxtlOqkDMBo5yXaIgpG9AetAyVNATdTid/jOxjo3TtIFKmJNoT9fdESkOth6FvOkOKfT3vjcX/vFaCwXk75TJOECSbLgoSYWNkj0Oxu1wBQzE0hDLFza0261NFGZrociYEd/7lRVI/Lbnl0sVtuVApz+LIkgNyRIrEJWekQq5JldQII4/kmbySN+vJerHerY9pa8aazeyTP7A+fwB+upmx</latexit>

basis set includes terms up to energy cutoff (see INCAR)1

2| ~G+ ~k|2 < Ecuto↵

<latexit sha1_base64="GYSFOMZm5le/t90Sj1+33Adwpz8=">AAACGXicbVDLSgNBEJz1GeNr1aOXxSAIQtgNARU8BET0GMFEIRvD7KRXh8w+mOkNhsn+hhd/xYsHRTzqyb9xEnPwVdB0UdXNTFeQCq7QdT+sqemZ2bn5wkJxcWl5ZdVeW2+qJJMMGiwRibwMqALBY2ggRwGXqQQaBQIugt7RyL/og1Q8ic9xkEI7otcxDzmjaKSO7fqhpEx7ua7kQ78PTJ/ku+Pey4dXlcPjjvYRblGzDJMwzPOOXXLL7hjOX+JNSIlMUO/Yb343YVkEMTJBlWp5boptTSVyJiAv+pmClLIevYaWoTGNQLX1+LLc2TZK1wkTaSpGZ6x+39A0UmoQBWYyonijfnsj8T+vlWG439Y8TjOEmH09FGbCwcQZxeR0uQSGYmAIZZKbvzrshpqo0IRZNCF4v0/+S5qVslctH5xVS7XqJI4C2SRbZId4ZI/UyCmpkwZh5I48kCfybN1bj9aL9fo1OmVNdjbID1jvn3gHodU=</latexit>

un

⇣~r,~k

⌘=

1

⌦1/2

X

~G

cn⇣~k, ~G

⌘expi

~G~r

<latexit sha1_base64="svJ+RhrQnrS7gUfmNN4BSOkKofo=">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</latexit>

WAVECAR (binary)

n

⇣~r,~k

⌘=

1

⌦1/2

X

~G

cn⇣~k, ~G

⌘expi(

~k+~G)~r

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k-grid sampling

10

integrating over BZ:

⇢ (~r) =1

⌦BZ

X

n

Z

BZfn

⇣~k⌘ �� n

⇣~r,~k

⌘��2d~k

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occupancy

convert to weighted sum over a discrete set of k-points:

⇢ (~r) =X

n~k

w⇣~k⌘fn

⇣~k⌘ �� n

⇣~r,~k

⌘��2d~k

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numerically, think trapezoidal method:Z b

af(x)dx ⇡

NX

k=1

f (xk�1) + f (xk)

2�xk

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solve f at particular x and sum

k-grid sampling: KPOINTS and IBZKPT files

~b1

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~b2

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BZ IBZ~k1

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~k2

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~k3

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set KPOINTS file1. auto schemes: Monkhorst-Pack vs. Gamma-centered

• equally-spaced mesh• MP does NOT include origin (Gamma)

• most symmetric point: outlier• saves resources• safer to choose Gamma-centered

• choose grid count (n) along each direction (even/odd)• add a shift

2. set your own points and weights

n~kio=

nui~b1 + vi~b2 + wi

~b3o

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{ui, vi, wi} =2i� n� 1

2n

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w~k1,~k2=

4

16=

1

4

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w~k3=

8

16=

1

2

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1

⌦BZ

Z

BZF⇣~k⌘d~k ! 1

4

⇣F⇣~k1

⌘+ F

⇣~k2

⌘⌘+

1

2F⇣~k3

⌘

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11

inspired byslide 10 of VASP Basics lecture

visual 2D example: 4 x 4 grid• 16 points total in BZ• 3 symmetrically-inequivalent ones

Meshes breaking symmetry

12

slide 12 of VASP Basics lecture

fcc, hex, rhlnon-uniform distribution of k-points

Gamma-centered is always safe!

sym⇣~k-mesh

⌘⌘ sym (BZ)

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60�

Pseudopotentials

13


• complicated, high-energy (very node-y) interactions at the core• combined valence+core requires too many plane wave terms• cannot be modeled beyond Li

• replace real core (up to rc) with effective potential• large rc converges quicker, less transferable (SOFTER)

Projector-Augmented Wave (PAW) Method

14

• addresses interaction between valence and core wave-functions• decomposition of the all-electron (full-potential) into three contributions• each has their “scope”: outside of which is modeled with smooth functions• combined with well-calculated “frozen core”

• this approach is among the most accurate for solids• among the top reasons to use VASP


exactpseudo (node-less)

plane wavespseudo-onsite

radial gridsexact onsiteradial grids

Exchange-correlation functional

15

local density approximation (LDA)

• : exchange and correlation per particle of a uniform electron gas of density

• tends to over-bind:• underestimates lattice parameters and volumes• overestimates cohesive energies, elastic (bulk) moduli, phonon frequencies

generalized gradient approximation (GGA)

• generally improves accuracy over LDA: can over-compensate• parameter-free GGA by Perdew-Wang (PW91)• simplified GGA by Perdew, Burke, Ernzerhof (PBE)

• standard for intermetallic solids

ELDAxc =

Z⇢ (~r) ✏xc (⇢ (~r)) d~r

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✏xc (⇢ (~r))

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⇢ (~r)

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EGGAxc =

Z⇢ (~r) ✏xc (⇢ (~r) ,r⇢ (~r)) d~r

<latexit sha1_base64="FR3mOhqcLSuigol1vVM5k2hLRUo=">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</latexit>

Exchange-correlation functional

16

meta-GGAs

hybrid functionals• incorporate a portion of exact-exchange from Hartree-Fock• 3-parameter functional by Becke, Lee, Yang, Parr (B3LYP)

• 3-parameters fitted from experimental data: empirical functional• standard for molecules

Emeta-GGAxc =

Z⇢ (~r) ✏xc

�⇢ (~r) ,r⇢ (~r) ,r2⇢ (~r)

�d~r

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Perdew’s Jacob’s ladder

17J. Phys. Chem. A 111, 10439 (2007)

Heavenchemical accuracy

fully non-local

hybrid meta-GGAhybrid GGA

meta-GGA

GGA

LDA

B1B95, BB1K, PBE1KCISB3LYP, B3P86, B3PW91, BH&LYP

BB95, MPW1K, TPSS, VSXC, SCAN

BLYP, BP86, BPW91, G96LYP,HCTH, OLYP, PBE

SPWL

EarthHartree-Fock theory

18

Example 1: Energy calculation of STD-CONV Alcd 01_vasp/1_ENERGY/1_Al/1_STDCONV

1. create POSCAR and KPOINTS file (10x10x10).2. run vasp

HINTS:• aluminum has an fcc structure• a = 4.05 angstroms (see aflow.org or Wikipedia)

19

Example 1: Energy calculation of STD-CONV Al

INCAR:

SYSTEM=AlIBRION=2 # relax with Conjugate GradientNSW=51 # relax for long, number of ionic stepsISIF=3 # relax appropriately, relax: forces, Stress

# tensor, positions, cell shape, cell volumeLORBIT=10 # get spin decompositionNBANDS=33 # KBIN::XVASP_INCAR_GetNBANDSISYM=2 # SYMMETRY=ONLWAVE=.FALSE. # WAVECAR=OFFLCHARG=.TRUE. # CHGCAR=ONPREC=Accurate # avoid wrap around errorsENMAX=336.42 # 1.4*ENMAX (240.3) of pseudopotentialsEDIFF=1E-6 # high accuracy requiredISMEAR=1 # for default (as metal)SIGMA=0.1 # for default (as metal)

20


POTCAR:

PAW_PBE Al 04Jan20013.00000000000000000parameters from PSCTR are:VRHFIN =Al: s2p1LEXCH = PEEATOM = 53.5387 eV, 3.9350 Ry

TITEL = PAW_PBE Al 04Jan2001LULTRA = F use ultrasoft PP ?IUNSCR = 1 unscreen: 0-lin 1-nonlin 2-noRPACOR = 1.500 partial core radiusPOMASS = 26.982; ZVAL = 3.000 mass and valenzRCORE = 1.900 outmost cutoff radiusRWIGS = 2.650; RWIGS = 1.402 wigner-seitz radiusENMAX = 240.300; ENMIN = 180.225 eVICORE = 2 local potentialLCOR = T correct aug chargesLPAW = T paw PP

functional

basis cutoff

21


POSCAR:

lattice vectors:Basis: 4 atoms• origin

•

•

•

~a1 = a~x, ~a2 = a~y, ~a3 = a~z

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1

2~a1 +

1

2~a2

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1

2~a1 +

1

2~a3

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1

2~a2 +

1

2~a3

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22


POSCAR:

Al1.000000

4.05000000000000 0.00000000000000 0.000000000000000.00000000000000 4.05000000000000 0.000000000000000.00000000000000 0.00000000000000 4.05000000000000

4Direct(4) [A4]

0.00000000000000 0.00000000000000 0.00000000000000 Al0.50000000000000 0.50000000000000 0.00000000000000 Al0.50000000000000 0.00000000000000 0.50000000000000 Al0.00000000000000 0.50000000000000 0.50000000000000 Al

a all along diagonal

Origin + 3 unique faces

23


KPOINTS:

KPOINTS File0 ! number of k-points = 0 ->automatic generation schemeGamma ! generate a Gamma centered grid10 10 10 ! subdivisions N_1, N_2 and N_3 along recipr. l. vectors0 0 0 ! optional shift of the mesh (s_1, s_2, s_3)

24


CONTCAR:Al/A_cF4_225_a.A params=4.05 SG=225 [ANR1.00000000000000

4.0418673187797358 0.0000000000000000 -0.00000000000000000.0000000000000000 4.0418673187797358 0.00000000000000000.0000000000000000 0.0000000000000000 4.0418673187797358

4Direct-0.0000000000000000 -0.0000000000000000 0.00000000000000000.5000000000000000 0.5000000000000000 0.00000000000000000.5000000000000000 0.0000000000000000 0.5000000000000000-0.0000000000000000 0.5000000000000000 0.5000000000000000

a changed

fractional coordinates did NOT change

25


OSZICAR:

N E dE d eps ncg rms rms(c)DAV: 1 0.568562714370E+01 0.56856E+01 -0.25715E+03 3704 0.191E+02DAV: 2 -0.145045587380E+02 -0.20190E+02 -0.19297E+02 4659 0.484E+01DAV: 3 -0.150420730511E+02 -0.53751E+00 -0.53093E+00 4658 0.601E+00DAV: 4 -0.150448253916E+02 -0.27523E-02 -0.27519E-02 4226 0.468E-01DAV: 5 -0.150448491042E+02 -0.23713E-04 -0.23712E-04 4687 0.419E-02 0.185E+00DAV: 6 -0.149932033785E+02 0.51646E-01 -0.46757E-03 4182 0.174E-01 0.115E+00DAV: 7 -0.149553889252E+02 0.37814E-01 -0.14850E-02 4281 0.290E-01 0.495E-02DAV: 8 -0.149554201153E+02 -0.31190E-04 -0.20305E-04 4456 0.407E-02 0.126E-02DAV: 9 -0.149554229353E+02 -0.28200E-05 -0.48230E-06 4451 0.517E-03 0.121E-03DAV: 10 -0.149554226525E+02 0.28280E-06 -0.74299E-08 3303 0.839E-04

1 F= -.14955423E+02 E0= -.14955211E+02 d E =-.149554E+02N E dE d eps ncg rms rms(c)

DAV: 1 -0.149560470184E+02 -0.14956E+02 -0.51405E-03 4062 0.181E-01 0.459E-02DAV: 2 -0.149560113004E+02 0.35718E-04 -0.93740E-05 4276 0.253E-02 0.286E-02DAV: 3 -0.149559846968E+02 0.26604E-04 -0.10835E-05 4108 0.688E-03 0.101E-03DAV: 4 -0.149559848928E+02 -0.19601E-06 -0.17982E-07 2933 0.130E-03

2 F= -.14955985E+02 E0= -.14955709E+02 d E =-.562240E-03N E dE d eps ncg rms rms(c)

DAV: 1 -0.149559970045E+02 -0.14956E+02 -0.71597E-05 4108 0.212E-02 0.541E-03DAV: 2 -0.149559962485E+02 0.75597E-06 -0.12187E-06 4165 0.299E-03

3 F= -.14955996E+02 E0= -.14955729E+02 d E =-.573596E-03

EDIFFG

26


OUTCAR:

…FREE ENERGIE OF THE ION-ELECTRON SYSTEM (eV)---------------------------------------------------free energy TOTEN = -14.955996 eV

energy without entropy= -14.955194 energy(sigma->0) = -14.955729

…POSITION TOTAL-FORCE (eV/Angst)…General timing and accounting informations for this job:========================================================

Total CPU time used (sec): 155.977User time (sec): 153.982

System time (sec): 1.996Elapsed time (sec): 156.168

energy for the cell

calculation time

zero forces (relaxed)

CONTCAR:

Al 1.00000000000000

4.0418673052840681 -0.0000000000000000 -0.00000000000000000.0000000000000000 4.0418673052840681 -0.0000000000000000

-0.0000000000000000 -0.0000000000000000 4.0418673052840681Al

4Direct-0.0000000000000000 -0.0000000000000000 0.00000000000000000.5000000000000000 0.5000000000000000 0.00000000000000000.5000000000000000 0.0000000000000000 0.50000000000000000.0000000000000000 0.5000000000000000 0.5000000000000000

0.00000000E+00 0.00000000E+00 0.00000000E+000.00000000E+00 0.00000000E+00 0.00000000E+000.00000000E+00 0.00000000E+00 0.00000000E+000.00000000E+00 0.00000000E+00 0.00000000E+00

27


lattice parameter changed

velocities

no change in direct coordinates

28

Exercise 1: Energy calculation of STD-PRIM Alcd ../2_STDPRIM

1. start with KPOINTS_10x10x10: create POSCAR for the standard primitive representation

2. In your small groups, also run:• KPOINTS_01x01x01• KPOINTS_02x02x02• KPOINTS_03x03x03• KPOINTS_04x04x04• KPOINTS_05x05x05• KPOINTS_15x15x15• KPOINTS_20x20x20

3. establish convergence of energy and lattice parameters with respect to the k-point grids

HINTS:

• a = 4.05 angstroms (see aflow.org or Wikipedia)

• lattice vectors:

• the standard primitive representation has 1 atom only (at the origin)

~a1 =a

2(~y + ~z) , ~a2 =

a

2(~z + ~x) , ~a3 =

a

2(~x+ ~y)

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29

Exercise 1: Energy calculation of STD-PRIM Al

OUTCAR:

…General timing and accounting informations for this job:========================================================

Total CPU time used (sec): 79.425User time (sec): 78.214

System time (sec): 1.212Elapsed time (sec): 79.458

calculation time cut in half

30

Exercise 1: Energy calculation of STD-PRIM Al

full comparison:

KPOINTS ENERGY(eV/atom) LAT_PARAM(\AA) TIME(sec)01 -0.130102 4.05 2.0002 -4.433748 3.97 4.8703 -3.160191 4.03 5.4604 -3.443518 4.05 7.3505 -3.680070 4.07 13.9110 -3.721916 4.03 46.9815 -3.743312 4.04 111.1820 -3.742644 4.04 239.02

-5

-4

-3

-2

-1

0

0 5 10 15 20

nkpt

E(e

V/at

om)

3.963.98

44.024.044.064.08

0 5 10 15 20

nkpt

a(a

ngst

rom

s)

050

100150200250300

0 5 10 15 20

nkpttim

e (s

ec)

31

Exercise 2: Energy calculation of Fecd ../../2_Fe

1. in your small groups, create the POSCARs for the different lattices and run vasp; do for both NO_SPIN and SPIN variants• for BCC: use experimental lattice parameter• for all other lattices, use experimental lattice parameter + 10%• HEX_1M: c/a = 0.9; HEX_1P: c/a = 1.1

2. compare the energies among the lattices, which is lower? Compare SPIN/NO_SPIN.3. compare original and relaxed lattice parameters

HINTS:• a = 2.87 angstroms (see aflow.org or Wikipedia)• the standard primitive representation has 1 atom only (at the origin)

• CUB lattice vectors:

• BCC lattice vectors:

• HEX lattice vectors:

~a1 = a~x, ~a2 = a~y, ~a3 = a~z

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~a1 =a

2(~y + ~z � ~x) , ~a2 =

a

2(~z + ~x� ~y) , ~a3 =

a

2(~x+ ~y � ~z)

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~a1 =a

2~x�

p3

2a~y, ~a2 =

a

2~x+

p3

2a~y, ~a3 = c~z

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32

INCAR (NO_SPIN):

SYSTEM=FeIBRION=2 # relax with Conjugate GradientNSW=51 # relax for long, number of ionic stepsISIF=3 # relax appropriately, relax: forces, Stress

# tensor, positions, cell shape, cell volumeLORBIT=10 # get spin decompositionNBANDS=45 # KBIN::XVASP_INCAR_GetNBANDSISYM=2 # SYMMETRY=ONLWAVE=.FALSE. # WAVECAR=OFFLCHARG=.TRUE. # CHGCAR=ONPREC=Accurate # avoid wrap around errorsENMAX=410.5332 # 1.4*ENMAX (293.238) of pseudopotentialsEDIFF=1E-6 # high accuracy requiredISMEAR=1 # for default (as metal)SIGMA=0.1 # for default (as metal)

Exercise 2: Energy calculation of Fe

33

INCAR (SPIN):

SYSTEM=FeIBRION=2 # relax with Conjugate GradientNSW=51 # relax for long, number of ionic stepsISIF=3 # relax appropriately, relax: forces, Stress

# tensor, positions, cell shape, cell volumeLORBIT=10 # get spin decompositionNBANDS=45 # KBIN::XVASP_INCAR_GetNBANDSISPIND=2 # SPIN=ON, enables spin-polarized calculationsISPIN=2 # SPIN=ON, turns on spin-polarized calculationMAGMOM= 1*1 # SPIN=ON - 1 atoms (default)ISYM=2 # SYMMETRY=ONLWAVE=.FALSE. # WAVECAR=OFFLCHARG=.TRUE. # CHGCAR=ONPREC=Accurate # avoid wrap around errorsENMAX=410.5332 # 1.4*ENMAX (293.238) of pseudopotentialsEDIFF=1E-6 # high accuracy requiredISMEAR=1 # for default (as metal)SIGMA=0.1 # for default (as metal)


spin ON settings

34

NO_SPIN:

STRUCTURE ENERGY(eV/atom) LAT_PARAM(\AA) TIME(sec)CUB -7.346038 2.28 152.63FCC -8.325992 3.45 18.14BCC -7.989485 2.76 77.20HEX_1M -7.759567 2.45 107.54HEX_1P -7.577295 2.45 463.80

SPIN:

STRUCTURE ENERGY(eV/atom) LAT_PARAM(\AA) TIME(sec)CUB -7.349056 2.28 272.36FCC -8.359111 3.45 69.68BCC -8.464997 2.83 114.79HEX_1M -7.803826 2.58 161.97HEX_1P -7.819675 2.66 702.71


without spin, Fe would be FCC

35

Exercise 3: DOS and band structure calculationscd ../../2_DOSBS

for all systems:1. perform DOS calculation: run vasp2. run DOS plotter: aflow --plotdos3. copy CHGCAR into BANDS calculation: cp CHGCAR ../2_BANDS4. create KPOINTS file consistent with the symmetry of the Bravais lattice5. run BANDS calculation: run vasp6. inside BANDS calculation, link OUTCAR from DOS calculation as OUTCAR.static

ln –sf ../1_DOS/OUTCAR ./OUTCAR.static1. inside BANDS calculation, link OUTCAR as OUTCAR.bands

ln –sf OUTCAR ./OUTCAR.bands1. run bandgap code: aflow --bandgap2. run band structure plotter: aflow --plotbanddos

HINTS:• for the BANDS calculation, see doi=10.1016/j.commatsci.2010.05.010:http://materials.duke.edu/auro/AUROARTICULA/j.commatsci.2010.05.010.pdf• Remember: alpha=ang(b,c), beta=ang(a,c), gamma=ang(a,b)

Let’s do example together: silicon

http://materials.duke.edu/auro/AUROARTICULA/j.commatsci.2010.05.010.pdf

36

INCAR (Si DOS):

SYSTEM=SiNBANDS=39 # KBIN::XVASP_INCAR_GetNBANDSISYM=2 # SYMMETRY=ONENMAX=343.483 # 1.4*ENMAX (245.345) of pseudopotentialsEDIFF=1E-6 # high accuracy requiredISMEAR=-5 # Performing RELAX_STATIC (tetrahedron method with Blochl correctionsSIGMA=0.05 # Performing RELAX_STATIC

# (so the DOS will not spill too much from band edges)NSW=0 # Performing RELAX_STATIC

# (zero ionic steps, so no relax, just static)IBRION=2 # Performing RELAX_STATIC (relax conj. grad, but single step

# so no relax)ISIF=2 # Performing RELAX_STATIC (relax ions, but single step so no relax)NELM=120 # Performing RELAX_STATIC (electronic steps max)NELMIN=2 # Performing RELAX_STATIC (electronic steps min)LORBIT=10 # Performing RELAX_STATIC (get spin decomposition)LCHARG=.TRUE. # Performing RELAX_STATICLAECHG=.TRUE. # Performing RELAX_STATIC (Bader ON)LWAVE=.FALSE. # Performing RELAX_STATICPREC=ACCURATE # Performing RELAX_STATIC (aleksey)EMIN= -30.0 # Performing RELAX_STATIC (aleksey) force search for EMIN (DOS E min)EMAX= 45.0 # Performing RELAX_STATIC (aleksey) (DOS E max)NEDOS= 5000 # Performing RELAX_STATIC (aleksey) (number of E steps)

Example 2: Si DOS and band structure calculations

print well-convergedCHGCAR for BANDS

37


0

1

2

3

4

�10 �5 0 5 10

spd

{{ i} , {Ei}}

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38

INCAR (Si BANDS):

SYSTEM=SiNBANDS=39 # KBIN::XVASP_INCAR_GetNBANDSISYM=2 # SYMMETRY=ONENMAX=343.483 # 1.4*ENMAX (245.345) of pseudopotentialsEDIFF=1E-6 # high accuracy requiredISMEAR=0 # Performing RELAX_STATIC_BANDS

# (put back Gauss, use 1 if metals)SIGMA=0.05 # Performing RELAX_STATIC_BANDS

# (so the DOS will not spill too much from band edges)NSW=0 # Performing RELAX_STATIC_BANDSIBRION=2 # Performing RELAX_STATIC_BANDSISIF=2 # Performing RELAX_STATIC_BANDSNELM=120 # Performing RELAX_STATIC_BANDSNELMIN=2 # Performing RELAX_STATIC_BANDSLORBIT=10 # Performing RELAX_STATIC_BANDSLCHARG=.FALSE. # Performing RELAX_STATIC_BANDSLAECHG=.TRUE. # Performing RELAX_STATIC_BANDS (Bader ON)LWAVE=.FALSE. # Performing RELAX_STATIC_BANDSPREC=ACCURATE # Performing RELAX_STATIC_BANDS (aleksey)ICHARG=11 # Performing RELAX_STATIC_BANDS (use CHGCAR.static)


use input CHGCAR from STATIC

39

KPOINTS (Si):

FCC (face-centered cubic) G-X-W-K-G-L-U-W-L-K U-X20 ! 20 gridsLine-modereciprocal


40

KPOINTS (Si BANDS):


0.0000 0.0000 0.0000 ! \Gamma0.5000 0.0000 0.5000 ! X

0.5000 0.0000 0.5000 ! X0.5000 0.2500 0.7500 ! W

0.5000 0.2500 0.7500 ! W0.3750 0.3750 0.7500 ! K

0.3750 0.3750 0.7500 ! K0.0000 0.0000 0.0000 ! \Gamma

0.0000 0.0000 0.0000 ! \Gamma0.5000 0.5000 0.5000 ! L

0.5000 0.5000 0.5000 ! L0.6250 0.2500 0.6250 ! U

0.6250 0.2500 0.6250 ! U0.5000 0.2500 0.7500 ! W

0.5000 0.2500 0.7500 ! W0.5000 0.5000 0.5000 ! L

0.5000 0.5000 0.5000 ! L0.3750 0.3750 0.7500 ! K

0.6250 0.2500 0.6250 ! U0.5000 0.0000 0.5000 ! X


41


�10

�5

0

5

10

� X W K � L U W L K U X 0 1 2 3 4

spd

{{ i} , {Ei}}

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42


Si:

System : SiSpin tag : 1Fermi level : +5.6321e+00

VBT CBB Egap Egap_fit TypeNet Result : -7.4000e-03 +6.0250e-01 +6.0990e-01 +1.7351e+00 insulator-indirect

correct typecompare to experiment (Google): does DFT over- or under-estimate?

43

Exercise 3: DOS and band structure calculationscd ../../2_DOSBS

for all systems:1. perform DOS calculation: run vasp2. run DOS plotter: aflow --plotdos3. copy CHGCAR into BANDS calculation: cp CHGCAR ../2_BANDS4. create KPOINTS file consistent with the symmetry of the Bravais lattice5. run BANDS calculation: run vasp6. inside BANDS calculation, link OUTCAR from DOS calculation as OUTCAR.static

ln –sf ../1_DOS/OUTCAR ./OUTCAR.static1. inside BANDS calculation, link OUTCAR as OUTCAR.bands

ln –sf OUTCAR ./OUTCAR.bands1. run bandgap code: aflow --bandgap2. run band structure plotter: aflow --plotbanddos

HINTS:• for the BANDS calculation, see doi=10.1016/j.commatsci.2010.05.010:

j.commatsci.2010.05.010.pdf• Remember: alpha=ang(b,c), beta=ang(a,c), gamma=ang(a,b)

44

KPOINTS (AsGa):



45

Exercise 3: DOS and band structure calculations

�10

�5

0

5

10

� X W K � L U W L K U X 0 1.5 3 4.5 6

spd

46


AsGa:

System : AsGa_hSpin tag : 1Fermi level : +4.1318e+00

VBT CBB Egap Egap_fit TypeNet Result : -3.9000e-03 +2.1400e-01 +2.1790e-01 +1.2067e+00 insulator-direct

correct typecompare to experiment (Google): does DFT over- or under-estimate?

47

KPOINTS (Ag):



48


�10

�5

0

5

10

� X W K � L U W L K U X 0 1.5 3 4.5 6

spd

49


Ag:

System : AgSpin tag : 1Fermi level : +3.1933e+00

VBT CBB Egap Egap_fit TypeNet Result : -1.0000e+00 -1.0000e+00 -1.0000e+09 -1.0000e+09 metal

50


KPOINTS (AsCo):

HEX (hexagonal) G-M-K-G-A-L-H-A L-M K-H20 ! 20 gridsLine-modereciprocal

51


�10

�5

0

5

10

� M K � A L H A L M K H 0 2.5 5 7.5 10

s

p

d

52


AsCo:

System : AsCoSpin tag : 1Fermi level : +6.7832e+00


53


KPOINTS (Fe):

BCC (body-centered cubic) G-H-N-G-P-H P-N20 ! 20 gridsLine-modereciprocal

54


�10

�5

0

5

10

� H N � P H P N �2 �1 0 1 2

s

p

d

55


Fe:

System : Fe_pvSpin tag : 2Fermi level : +5.3075e+00

VBT CBB Egap Egap_fit TypeMajority Spin : -1.0000e+00 -1.0000e+00 -1.0000e+09 -1.0000e+09 metalMinority Spin : -1.0000e+00 -1.0000e+00 -1.0000e+09 -1.0000e+09 metalNet Result : -1.0000e+00 -1.0000e+00 -1.0000e+09 -1.0000e+09 metal

spin-polarized result

56


KPOINTS (As):

RHL1 (rhombohedral alpha < 90) G-L-B1 B-Z-G-X Q-F-P1-Z L-P20 ! 20 gridsLine-modereciprocal

57


�10

�5

0

5

10

� L B1 B Z � X Q F P 1 Z L P 0 1 2 3 4

spd

58


As:

System : AsSpin tag : 1Fermi level : +4.8813e+00


59


KPOINTS (Sb):

RHL1 (rhombohedral alpha < 90) G-L-B1 B-Z-G-X Q-F-P1-Z L-P20 ! 20 gridsLine-modereciprocal

60


�10

�5

0

5

10

� L B1 B Z � X Q F P 1 Z L P 0 1 2 3 4

spd

61


Sb:

System : SbSpin tag : 1Fermi level : +6.5861e+00


62


KPOINTS (Ge):

ORCI (body-centered orthorhombc a < b < c) G-X-L-T-W-R-X1-Z-G-Y-S-W L1-Y Y1-Z20 ! 20 gridsLine-modereciprocal

63


�10

�5

0

5

10

� XL T W R X1 Z � Y S W L1 Y Y 1Z 0 1 2 3 4

spd

64


Ge:

System : Ge_hSpin tag : 1Fermi level : +7.5438e+00


65

References

1. VASP Basics lecture: https://www.vasp.at/wiki/images/5/5d/VASP_lecture_Basics.pdf2. Koch and Holthausen, “A Chemist’s Guide to Density Functional Theory”3. Sholl and Steckel, “Density Functional Theory: A Practical Introduction”4. Parr and Yang, “Density-Functional Theory of Atoms and Molecules”5. Griffiths, “Introduction to Quantum Mechanics”6. Ratner and Schatz, “Introduction to Quantum Mechanics in Chemistry”7. Szabo and Ostlund, “Modern Quantum Chemistry: Introduction to Advanced Electronic

Structure Theory”8. Bader, “Atoms in Molecules: A Quantum Theory”9. Matta and Boyd, “The Quantum Theory of Atoms in Molecules: From Solid State to DNA

and Drug Design”, “Chapter 1.1: An Introduction to the Quantum Theory of Atoms in Molecules”

Introduction to DFT: VASP - Center for Autonomous Materials ...

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