2221-5 hands-on tutorial on electronic structure...
TRANSCRIPT
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2221-5
Hands-on Tutorial on Electronic Structure Computations
Shobhana Narasimhan
17 - 21 January 2011
Jawaharlal Nehru Centre for Advanced Scientific Research Banglalore
India
The "Self Consistent Field" SCF) Loop and Some Relevant Input Parameters for Quantum--ESPRESSO
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����������� � ���������������������
������������ ����� ������ ��
�������� ����������
Shobhana NarasimhanTheoretical Sciences Unit
JNCASR, [email protected]
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������������������������ �8
!��� �����"�#������������ ����$$$
� Are you an experienced Quantum Espresso user?
Yes
� Are you an extraordinarily nice person?
No.
PLEASE GO AWAY!
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������������������������ �10
!��� �����"�#������������ ����$$$
� Are you an experienced Quantum Espresso user?
Yes
� Are you an extraordinarily nice person?
Yes
It would be really nice if you could stick around here andhelp with the hands-�����������
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������������������������ �11
!��� �����"�#������������ ����$$$
� Are you an experienced Quantum Espresso user?
No.
Then this session is aimed at YOU!(But we are assuming that you are familiar withelectronic structure calculations, linux commands, etc., etc.)
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������������������������ �12
%��&���������������
� Want to solve the Kohn-Sham equations:
� Note that self-consistent solution necessary, as Hdepends on solution:
� Convention:
)()()]([)]([)(21 2 rrrrr iiiX CHnuc nVnVV ��� ���
�� ����
H
Hrni �� )(}{�
1��� eme �
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������������������������ �13
����'����� � �� �� ������� ���
Vnuc �������������
�������������n(r)
���������VH[n] ��VXC[n]
Veff�r���Vnuc(r) + VH (r) + VXC (r)
H�i(r) = [-1/2�2 ��Veff(r)] �i(r) ���i �i(r)
�������������n(r) = �i|�i(r)|2
��������������
����� ����!� "����������������������#$������$���%
&����������n(r)
'���
(�
How to solve theKohn-Sham eqns.for a set of fixednuclear (ionic)positions.
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������������������������ �14
�����(����)�������'��#� ���
� For a periodic system:
� The plane waves that appear in this expansion can be represented as a grid in k-space:
rGk
GGkk r �����
)(,
1)( iec�where G = reciprocallattice vector
kx
ky � Only true for periodic systems that grid is discrete.� In principle, still need
infinite number of plane waves.
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������������������������ �15
%���'� ��*� �������(����+�������
� In practice, the contribution from higher Fourier components (large |k+G |) is small.
� So truncate the expansion at some value of |k+G |.� Traditional to express this cut-off in energy units:
cu tEm
��
2|| 22 Gk�
kx
E=Ecut
ky
Input parameter �������
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������������������������ �16
��',��*������*�'�-� �'� -�'�
� Must always check.� Monotonic (variational).
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� ��./�0�����*� ��1������'2��#� �
�������������
������������������������ �
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������������������������ �18
3�-� ����'��#� ���#� �
All periodic systems can be specified by a Bravais Lattice and an atomic basis.
+ =
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������������������������ �19
3�-� ����'��#� ��4�������5� �'�6�7�� ����
- Gives the type of Bravais lattice (SC, BCC, Hex, etc.)
Input parameter ����
Input parameters {������� �}- Give the lengths [& directions, if necessary] of the lattice vectors a1, a2, a3
Note that one can choose a non-primitive unit cell
(e.g., 4 atom SC cell for FCC structure).
a1
a2
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������������������������ �20
����(� ����7�� ������ 3�-����#��-��8
Input parameter ���
- ����� ��������������� ������� ��� ���������������������-� ������������������������������ ��������������������������������� ����� ����������������������������������������
- Number of atoms in the unit cell
Input parameter ����- Number of types of atoms
FIELD � �!"#$ �!! %�
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������������������������ �21
� ��9/��� �����*�Vnuc
Vnuc �������������
�������������n(r)
���������VH[n] ��VXC[n]
Veff�r���Vnuc(r) + VH (r) + VXC (r)
H�i(r) = [-1/2�2 ��Veff(r)] �i(r) ���i �i(r)
�������������n(r) = �i|�i(r)|2
��������������
����� ����!� "����������������������#$������$���%
&����������n(r)
'���
(�
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������������������������ �22
��'������ � ���
Electrons experience a Coulomb potential due to the nuclei.
This has a known and simple form:
But this leads to computational problems!
rZ
Vnuc �
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������������������������ �23
����������������(���4����
Core wavefunctions:sharply peaked near nucleus.
High Fourier components present
i.e., need large Ecut �
Valence wavefunctions:lots of wiggles near nucleus.
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������������������������ �24
���� ��������������(���4����
Core wavefunctions:sharply peaked near nucleus.
High Fourier components present
i.e., need large Ecut �
Valence wavefunctions:lots of wiggles near nucleus.
"��#����������������core electrons!
Remove wiggles from valence electrons.
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������������������������ �25
������� � ����
Replace nuclear potential by pseudopotential
This is a numerical trick that solves these problems
There are different kinds of pseudopotentials (Norm conserving pseudopotentials, ultrasoft pseudopotentials, etc.)
Which kind you use depends on the element.
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������������������������ �26
������� � ������������� ������������ 9
Go to http://www.quantum-�����������$�����%�����'*+0"1�
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������������������������ �27
������� � ������������� ������������ :
Click on element for which pseudopotential wanted.
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������������������������ �28
������� � ������������� ����������
'������������ �#��� ���gives information about :
� type of exchange-correlation functional
� type of pseudopotential
� e.g.:
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������������������������ �29
���� �)�Vion �������� ����������
� �!"#�$�"!��&�'()*+),*'&�+���-���-���+.$/ '0*+12*' +���-��-���#�3+.$/ ''(4+555' +���-���#�3+.$/
� ���������������� depend on type of pseudopotentials used (should test).
When using ultrasoft pseudopotentials, set ������� = 8-12 � ������� !!
e.g, for calculation on BaTiO3:
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������������������������ �30
���� �)�Vion �������� ����������
Should have same exchange-correlation functional for all pseudopotentials.
input
output
oops!
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������������������������ �31
� ��:/���� ����;��������n(r)
Vion �������������
�������������n(r)
���������VH[n] ��VXC[n]
Veff�r���Vion(r) + VH (r) + VXC (r)
H�i(r) = [-1/2�2 ��Veff(r)] �i(r) ���i �i(r)
�������������n(r) = �i|�i(r)|2
��������������
����� ����!� "����������������������#$������$���%
&����������n(r)
'���
(�
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������������������������ �32
� �� ��*�(�����' ����
���������'��� ��������� �������
Superposition of atomic orbitals
The closer your starting wavefunction is to the true � �����������������5����������5���������������������#��necessarily know to start with!), the fewer the scf iterations needed.
�6���7��������������������������� ���� �������������%��- Plato
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������������������������ �33
� ���<�)�=/����' ����� � ���
Vion �������������
�������������n(r) or �i(r)
���������VH[n] ��VXC[n]
Veff�r���Vnuc(r) + VH (r) + VXC (r)
H�i(r) = [-1/2�2 ��Veff(r)] �i(r) ���i �i(r)
�������������n(r) = �i|�i(r)|2
��������������
����� ����!� "����������������������#$������$���%
&����������n(r)
'���
(�
Note that typeof exchange-correlation chosen while specifying pseudopotential
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������������������������ �34
�+'���*������� ������ � ���
VXC � �EXC/�n contains all the many-body information. Known [numerically, from Quantum Monte Carlo ;
various analytical approximations] for homogeneous electron gas.
Local Density Approximation:
Exc[n] = � n(r) VxcHOM[n(r)] dr
-surprisingly successful!
Generalized Gradient Approximation(s): Include terms involving gradients of n(r)
Replace
�6
��5(�����
(in name of pseudopotential)
(in name of pseudopotential)
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������������������������ �35
� ��>/�0��*�����?� ���
Vion �������������
�������������n(r)
���������VH[n] ��VXC[n]
Veff�r���Vion(r) + VH (r) + VXC (r)
H�i(r) = [-1/2�2 ��Veff(r)] �i(r) ���i �i(r)
�������������n(r) = �i|�i(r)|2
��������������
����� ����!� "����������������������#$������$���%
&����������n(r)
'���
(�
Expensive! �
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������������������������ �36
0��*�����?� ���
Need to diagonalize a matrix of size NPW � NPW
NPW >> Nb = number of bands required = Ne/2 or a little more (for metals).
OK to obtain lowest few eigenvalues. Exact diagonalization is expensive! Use iterative diagonalizers that recast
diagonalization as a minimization problem.Input parameter � ������ 6�� ��
Input parameter ����-which algorithm used for iterative diagonalization
-how many eigenvalues computedfor metals, choose depending on value of �������
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������������������������ �37
� ��@/��-�����*�0��� #
Vion �������������
�������������n(r)
���������VH[n] ��VXC[n]
Veff�r���Vion(r) + VH (r) + VXC (r)
H�i(r) = [-1/2�2 ��Veff(r)] �i(r) ���i �i(r)
�������������n(r) = �i|�i(r)|2
��������������
����� ����!� "����������������������#$������$���%
&����������n(r)
'���
(�
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������������������������ �38
04���������A�������
� Many quantities (e.g., n, E tot) involve sums over k .� In principle, need infinite number of k#�� In practice, sum over a finite number: 9<��* �������� Number needed depends on band structure.� Typically need more k#��������� ��� Need to test convergence wrt k-point sampling.
k
� �F ��
�BZ
wPN
Pk
kk
k )(1
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������������������������ �39
0
nk1, nk2, nk3, k1, k2, k3
K_POINTS { tpiba | automatic | crystal | gamma }
%#������,����� �����
� Special Points: [Chadi & Cohen]Points designed to give quick convergence for particular crystal structures.
� Monkhorst-Pack:Equally spaced mesh in reciprocal space.> ��7��������������������XZ���-�������#\��������XZ�������#\
b1
1st BZb2
���Z ���� ���#5�����>-P mesh:
nk1=nk2=4
shift
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40
�����*�'�-� �4A��������*
Note: Differences in energy usually converge faster than absolute value of total energy because of error cancellation (if supercells & k-points are identical or commensurate).
Madhura Marathe
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41
��������-� ��B ���
Recall:
For metals, at T=0, this corresponds to (for highest band) an integral over all wave-vectors contained within the Fermi surface, i.e., for highest band, sharp discontinuity in k-space between occupied and unoccupied �� ���������many k-points to reproduce this accurately.
Also can lead to scf convergence problemsbecause of band-crossings above/below Fermi level.
� ��
�occn BZ
n kdPP 33 )(
)2(k
�
Fermi Surface of Cuiramis.cea.fr
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42
������������*�C
Problems arise because of sharp discontinuity at Fermi surface / Fermi energy.
�*�� �� this out using a smooth operator!
Will now converge faster w.r.t. number of k-points (but not necessarily to the right answer!)
The larger the smearing, the quicker the convergence w.r.t. number of k-points, but the greater the error introduced.
The trick is to use a clever smearing function so that the error is small.
PhD Comics
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43
������*�������� ����������
������� ��'����������
���������� ��������������������������������������������������
Instruction: use smearing
Type ofsmearing
������� Smearing width
Methfessel & Paxton, Phys. Rev. B 40, 3616 (1989).Marzari & Vanderbilt, Phys Rev. Lett. 82, 3296 (1999).
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44
�����*�'�-� �*����)�������*
Gaussian: Methfessel-Paxton:
represents an energy difference of 1 mRy
Madhura Marathe
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45
�����*�'�-� �,����� ��)�������*�-�� �
R. Gebauer
e.g., for bcc Fe, using 14 �14 � 14 grid:
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������������������������ �46
� ��D/���',���������*�'� '����
Vion �������������
�������������n(r) or �i(r)
���������VH[n] ��VXC[n]
Veff�r���Vnuc(r) + VH (r) + VXC (r)
H�i(r) = [-1/2�2 ��Veff(r)] �i(r) ���i �i(r)
�������������n(r) = �i|�i(r)|2
��������������
����� ����!� "����������������������#$������$���%
&����������n(r)
'���
(�
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������������������������ �47
%� ��*������'��'����*�'
Compare nth and (n-1)th approximations for density, and see if they are close enough that self-consistency has been achieved.
Examine squared norm of difference between the �� ���������������������������������� �������������be close to zero.
Input parameter ����#���
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������������������������ �48
� ��E/�B�+��*
Vion �������������
�������������n(r) or �i(r)
���������VH[n] ��VXC[n]
Veff�r���Vnuc(r) + VH (r) + VXC (r)
H�i(r) = [-1/2�2 ��Veff(r)] �i(r) ���i �i(r)
�������������n(r) = �i|�i(r)|2
��������������
����� ����!� "����������������������#$������$���%
&����������n(r)
'���
(�
Can take a longtime to reachself-consistency!
�
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������������������������ �49
B�+��*
� Iterations n of self-consistent cycle:
� Successive approximations to density:
nin(n) � nout(n) � nin(n+1).
� nout(n) fed directly as nin(n+1) }}�~�5���� ���������#��converge.
� Need to mix, take some combination of input and output densities (may include information from several previous iterations).
� Goal is to achieve self consistency (nout = nin ) in as few iterations as possible.
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������������������������ �50
B�+��*�������� ����������
Input parameter � � ��#����
Input parameter � � ��#����
-How much of new density is used at each step-Typically use value between 0.1 & 0.7
-Prescription used for mixing.
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������������������������ �51
�� �� ����� � ��/�%� ������*#
� Perhaps the most important output quantity is the TOTAL ENERGY
� Can use, e.g., to optimize structure
� e.g., for a cubic crystal, where the structure can be specified by a single parameter (side of cube):