spin polarized version of dft & accuracy studies of dft

CCOORRNNEELLLL U N I V E R S I T Y

MAE 715 – Atomistic Modeling of Materials

N. Zabaras (3/5/2012) 1

Material for these notes follows closely a lecture provided by Prof. G. Ceder

for MIT’s Atomistic Materias Moldeling course.

MIT’s 3.320 course notes (Prof. G. Ceder)

SPIN POLARIZED VERSION OF DFT

& ACCURACY STUDIES OF DFT

http://ocw.mit.edu/OcwWeb/Materials-Science-and-Engineering/3-320Spring-2005/CourseHome/index.htm



N. Zabaras (3/5/2012) 2


In the Hohenberg-Kohn theorem, everything is expressed in terms of the charge

density and the electron spin does not appear explicitly. Electrons have

spin and if we don’t consider any coupling with the angular momentum is just

up or down spin (+) (-) ½ mB , where mB is the Bohr magnetron).

We’ll treat here spin as a scalar quantity so spin is +1/2 or –1/2 in the

appropriate units.

In reality when spin it couples with the angular momentum becomes a vector

quantity. Also when it couples to an external magnetic field it becomes a vector

quantity. But most times that you will use spin you will simply treated it as a

scalar. Spins up and down are often shown with an either up or down arrow.

We need to deal with the spin because of the Pauli exclusion principle -- two

electrons cannot be in exactly the same quantum state.



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That means that if you have 2 up electrons approaching each other versus an up

and a down electron, these will approach each other differently.

The 2 electrons shown above are in the same spin state so if you bring them

very close, they get the same coordinate as well so they have the same

quantum numbers and the Pauli exclusion principle prevents that. So the

Pauli exclusion principle keeps electrons with parallel spin away from

each other.

On the other hand, if you have electrons with anti-parallel spin, even if these

are at the same position, they don’t have the same spin, so they don’t have the

same quantum numbers so the Pauli exclusion principle is not violated.

The Pauli exclusion principle keeps the electrons away without an explicit

term for it in the Hamiltonian. The spin contribution comes from anti-

symmetrizing the wave function.



N. Zabaras (3/5/2012) 4

HUND’S RULES

Consequences of the Pauli exclusion principle is Hund’s rules. For example, if

you add electrons to e.g. 5 d levels, you are going to add them first with

parallel spin because the Pauli exclusion principle is satisfied and then you will

start filling them with anti-parallel levels.

The same is true with solids. If let’s say 5 d levels remain degenerate (at the

same level or with the split not too large), you are going to fill them according

to Hund’s rule:

In this case you have a strong magnetic moment (5 electron with spins up with

no down spin).

Strong magnetic moment



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To fill these levels with 5 electrons with parallel spin you would have to put them

on the top two separated levels but these levels are so much higher.

Instead, you can pay the Hund’s rule penalty and put the 4th and 5th electrons on

a lower orbital with anti-parallel spin.

On the other hand, let’s say, you are in an environment that splits off 2 levels.

Filling of different

orbitals gives the atom

different chemical properties Net spin (magnetic moment)

Levels split



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Whether you get a lot of magnetic spin and magnetic moment, depends on how

much your orbital will split in the end.

This can have significant consequences on the physical properties of your

material.

So in the DFT calculation you will carry a magnetic moment locally when the up-

density and the down-density are not the same.

In our DFT calculations, the energy and the potential are functions of the charge

density. So the charge density somehow needs to contain the information

about electron spin and magnetic moment.



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LOCAL SPIN DENSITY APPROXIMATION (LSDA)

In principle Exc[ρ] “knows” about this effect, but in practice it is poorly

approximated since only the total charge density is a variable.

For a given charge density there would probably be a certain amount of spin

polarization. This should be in the functional Exc[ρ].

We treat the up and the down densities separately. So you do the local spin

density approximation which is the spin polarized version of the local density

approximation (LSDA). There is also a spin-polarized version of GGA.

You have a separate density for the up electrons and a separate density for the

down electrons and the two will interact differently, i.e. the down will interact

differently with up than the up with down. This comes from the way the

exchange correlation potentials are defined.



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LOCAL SPIN DENSITY APPROXIMATION (LSDA)

The idea is similar to restricted and unrestricted Hartree-Fock.

If you have spin polarized materials it’s often much more useful to look at

spin densities than at charge densities. Recall from a figure in an earlier

lecture that you don’t see where the electrons are or any fine structure of

bonding when you just look at charge densities.



N. Zabaras (3/5/2012) 9

Plots of charge density

You see a lots of intensity from the Oxygen layers -- valence electrons only (the

O2-). Co has somewhat less valence electrons on it and you see less intensity.

Li is ionized to Li+ so it has no valence electrons and you see almost nothing.

Here is an example of

Lithium Cobalt Oxide

(LiCoO2) , a transition

metal oxide.

Its a layered material,

with layers of Oxygen

(red), layers of Cobalt

(black) and layers of

Lithium (yellow).

If you plot the charge

density in a plane you

only see big blobs. red is the neutral color in these figures –

i.e. its zero

MIT’s 3.320




N. Zabaras (3/5/2012) 10

Spin polarization density

This Figure of charge density does not give you a lots of detail. If you rather plot

the spin polarization density (up minus down), this will indicate how much

magnetic moment there is locally.

MIT’s 3.320




N. Zabaras (3/5/2012) 11

SPIN POLARIZATION DENSITY

The Figure here is similar to the earlier one but with Nickel and Manganese ions

in it.

Note that you do not see the oxygen at all since oxygen doesn’t have spin (it’s a

filled shell, every time you have filled shells, you don’t have spin).

Plots of spin density often allow you to filter certain ions. For example, the

transitional metals don’t have spin on them and you see that clearly. This is a

useful trick that you should keep in mind.

Oxygen

MIT’s 3.320




N. Zabaras (3/5/2012) 12


Spin polarization can be important (in

particular for transition metal compounds)

If transition metal oxides, most of the time you’ll get the structure right but things

get more subtle. A transition metal has local d states that often have

significant spin polarization. So you need to turn on spin polarization.

It often gets worse. Remember that your spin is a scalar (up or down). Now you

have a spatial degree of freedom of how to organize that spin on the ions. If

you have a bunch of ions, you could put them all with the same direction, that’s a

ferromagnet. Or you could put them with alternating directions as a kind of anti-

ferromagnet.

There are many ways to make them anti-ferromagnetic. And unfortunately in

transition metal oxides it often matters because there is not only a strong spin

polarization effect on the energy but there’s a strong effect of the interaction

between spins on different ions. Lets see an example next.



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If you do a non spin-polarized calculation, you get a huge error, C2/m is lower in

energy by 250 meV. If you turn on spin-polarization but make them

ferromagnetic, they are degenerate. If you make them anti-ferromagnetic,

Pmmm is the lowest in energy.

This is the

structure of

Lithium

Manganese

Oxide (LiMnO2)

MIT’s 3.320

It is shown here a comparison

between 2 structures labeled by

their symmetry. One is C2/m and

one is Pmmn.

There are rather similar structures

but one is orthorombic and the

other is monoclinic. The correct

real crystal structure is Pmmn.




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If you take this material at room temperature, it’s paramagnetic and thus you

may think that it does not have any spin polarization, since there’s no net

moment. That’s wrong because a paramagnet still has a local moment, the

ions still have a moment on them that is just randomly oriented. So you still

need to represent that local moment. This is a very common mistake in these

type of calculations.

MIT’s 3.320




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It turns out that’s the biggest effect on the energy is the fact that you have that

local moment. It’s not necessarily how they are arranged. How they are

arranged makes you go from the difference between the 2 structures in the Ferro

to the difference in the A-Ferro configuration (which are small). However, by

turning on the local moment makes you go from the difference in the NSP to the

difference e.g. in A-Ferro (which is large).

It is only non-magnetic

materials or diamagnetic

materials for which

you don’t really need

spin polarization.

MIT’s 3.320




N. Zabaras (3/5/2012) 16

Orbital filling depends on spin polarization

This effect is important because if you spin-polarize an ion you fill different

orbitals. This is shown below for 5 d-orbitals.

This is typically how they split in most oxides. Every time an ion is octahedral,

5d orbitals tend to split as 3 down-2 up, in some cases 2 down-3 up.

MIT’s 3.320




N. Zabaras (3/5/2012) 17

Orbital filling depends on spin polarization

By spin polarizing you create essentially a different ion. It’s not really an issue of

magnetism because magnetic effects seem to be small in materials. But it’s the

fact that you create chemically a different ion because you fill different levels.

That’s really why these energy differences are so big.

Let’s say you have to put 4

electrons in there. You can put

them on what’s called high

spin or in low spin with no

moment. These 2 ions have

different chemical properties

because the electrons occupy

different d-orbitals. The high

spin orbital points in a

different direction than the

low-spin orbital.

MIT’s 3.320




N. Zabaras (3/5/2012) 18

NUMERICAL ACCURACY: Atomic energies

With DFT, we calculate simple things (charge densities, band structures and

energies). Other properties (e.g. corrosion require many steps).

Let us start with the energies of the atoms. LDA underestimates the stability

of an atom, GGA is a bit closer.

MIT’s 3.320

The numbers in this Table

should have a (–) sign in

front of them (summing

all the electronic states in

the atom gives binding)




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ENERGIES OF ATOMS

The Table provides the experimental numbers, the LDA number and then the

GGA with a fairly recent implementation of the exchange correlation function.

There is a very systematic trend in the data. In the LDA, in the atoms, the

electrons are not bound enough, in the GGA they are somewhat closer to

the experiment.

MIT’s 3.320




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BINDING ENERGY FOR SMALL MOLECULES

When you look at small molecules, the binding energy is too high in LDA,

GGA is closer but sometimes the bound too weak. Hartree-Fock is way off.

Here we compute the binding energy for diatomic

molecules, i.e. the energy to pull them apart. We

have a molecule AB with r the vector between them.

We are looking at the well depth in order to

compute the binding energy.

MIT’s 3.320




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ERROR CANCELLATION IN DFT

Suppose you are interested in an oxidation reaction and you want to calculate

the state of an oxide and compare the chemical potential of the Oxygen there to

that of Oxygen gas.

Aluminum + Oxygen Aluminum Oxide

We just showed that there is a big error in the Oxygen gas. How much of that

error carries over to the solid?

If we make exactly the same error in the solid, then the reaction energy is

perfect because we subtract the two.

A lot of practical things you do with density functional theory depend on error

cancellation. You will have more error cancellation as the states that you

subtract are more physically similar.



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BINDING ENERGY FOR SMALL MOLECULES

As an example, for H2 the LDA binding is too strong (-4.913). With the GGA,

the binding is too weak (-4.540). The experimental value is -4.753 eV.

Uncorrected Hartree-Fock in also shown here.

For O2, the experimental binding energy is only –5.2 ev. LDA binds it by -7.5 ev

(more than 2 ev off). GGA gets you a little closer. Uncorrected Hartree-Fock is

off.

Uncorrected Hartree-Fock should not be used at all as the binding energy is

way off.

MIT’s 3.320




N. Zabaras (3/5/2012) 23


But let’s say that you look at oxidation of a metal:

Aluminum + Oxygen Aluminum Oxide

The Oxygen in Aluminum Oxide is very different from the Oxygen in the O2

molecule so not all the error will cancel.

Let’s say that we had a 2 ev error in the molecule, this is 1 ev per Oxygen.

That’s an enormous effect.

Note that the chemical potential m (which relates directly to the energy) is given

in terms of the partial pressure of Oxygen as: m = mo + RT log P.

If e.g. you want to calculate the partial pressure of Oxygen at which something

oxidizes, the Oxygen pressure goes exponentially with the energetics. So, if

you have a 1 ev error at room temperature, your error in the Oxygen pressure is

the exponential of 1 ev/RT, which is quite high.



N. Zabaras (3/5/2012) 24


When we look at reactions between solids most of the error tends to cancel and

we get much better accuracy. Note that 1eV error in our reactions energy is

very significant.

The point to emphasize again is that we get less error cancellation as the states

we are comparing are different, e.g. going from a gas to a solid is a significant

difference.



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Lattice parameters in solids

The GGA results are always bigger. GGA underbinds but generally the GGA

error is more variable – less predictable. In metals you tend to be most of the time

on the high side.

An LDA lattice parameter that agrees with the experiment is usually based on a

wrong calculation!

Lattice parameter prediction for

solids is consistent with what we

see in molecules. The LDA

values are almost always

smaller than the experimental

predictions (by a factor of

1%~2%).

LDA overbinds.

MIT’s 3.320




N. Zabaras (3/5/2012) 26

BULK MODULUS IN SOLIDS

In almost all cases (except Si), the LDA bulk modulus is too large, so the material

is too stiff. Recall that LDA over-binds. LDA tends to be too stiff. GGA is too soft.

Bulk Modulus in Solids (in GPA)

Remember that when the bonding energy is too high, the lattice parameter is too

small and the material is too stiff.

Bulk modulus effects will also transfer into vibrational frequencies. In materials

when you are too stiff, you’ll have higher vibrational frequecies.

MIT’s 3.320




N. Zabaras (3/5/2012) 27

LATTICE PARAMETERS IN OXIDES

Here are the lattice parameters for oxides. Note that the errors tend to be

slightly larger.

MIT’s 3.320




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SUMMARY OF GEOMETRY PREDICTION

LDA under-predicts bond lengths

GGA error is less systematic though over-prediction is common.

errors are in many cases (e.g. semi-conductor metals) < 1%,

for transition metal oxides < 5%.



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STRUCTURE PREDICTION AND REQUIRED ENERGY

You often want to know if you put energy in a particular arrangement is that

lower in energy than some other arrangement? This way you can predict what

the most stable structure. This could be applicable for crystal structure or

surfaces.

For Vanadium, the atomic energy (the energy of all its electrons and not just the

valence electrons) in Rydbergs is given below assuming FCC and BCC

configurations.

The first line is the atom (not in a solid),

the 2nd line is the fcc ion, and the 3nd

line is the bcc ion.

Look at the differences. If you are going from the atom to the solid, your first

4 digits don’t even change!

The Fcc-Bcc difference is 0.02 Ry,

only 0.001% of the total energy.

In many cases we are going to work with

energy differences that are 10-6 or 10-7

times the total energy. This is telling you

how much numerical accuracy we need.

Mixing energies are also of order 10-6

fraction of the total E



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COHESIVE ENERGY

Cohesive energy for V is 0.638 Ry (0.03% of total E)

The cohesive energy is only 0.03% of the total energy. If you are calculating the

cohesive energy by first calculating the total energy of a solid and then

calculating the atomic energy, you will need to do these things very accurately

because you are going to subtract them and most of what you subtract is nearly

the same.

To get any significance in the results, you need to have high numerical

accuracy.



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MIXING ENERGIES

Mixing energies are also of order of 10-6 fraction of the total E

Suppose you mix Vanadium with Platinum and you want to know what is the

mixing entropy (that sets the whole temperature scale for mixing and the phase

diagram topology). The mixing energy tends to be a 10-6 to 10-7 times of the

total energy.

Unfortunately, the total energy is not accurate up to a fraction of 10-6. The only

reason we can get physical behavior right is because a lot of the error DFT

makes in LDA and GGA is systematic. A lots of it cancels away when you take

energy differences.

When we do fcc and bcc Vanadium, we may have an error of 10-4 in the energy

but most of it cancels away when we take the energy differences. The less

cancellation we have, the bigger the error on the result.



N. Zabaras (3/5/2012) 32

ELEMENT CRYSTAL STRUCTURE PREDICTIONS: bcc or fcc?

The comparison here is done in GGA with standard pseudopotential method. In

red we have metals that are experimentally fcc and in green we have metals

that are experimentally bcc.

We show the calculated

energy difference

between bcc and fcc.

When that is positive the

bcc energy is higher than

fcc so fcc is preferred

over bcc. If it’s negative

then bcc is preferred over

fcc.

All results are correct.

http://burgaz.mit.edu/PUBLICATIONS/PDF/Calphad.pdf




N. Zabaras (3/5/2012) 33

ELEMENT CRYSTAL STRUCTURE PREDICTIONS: hcp or fcc?

You could do a more

subtle comparison,

look at the difference

between hcp and

fcc. hcp and fcc are

much more alike --

they are all close

packed. It’s just a

difference in

stacking: abab

versus abcabc.

Again you see that all results are correct. The red ones are the fcc ones

(first line is positive). The yellow ones have the 1st number negative.




N. Zabaras (3/5/2012) 34

REACTION ENERGIES

Lets discuss reaction energies.

For example, a metal Lithium bcc with another metal Aluminum forming a Li-Al

compound. The LDA is 10% of from the experimental reaction energy. Metallic

reaction energies are somewhere in the range 5 to 15%.

The one that is next is Copper-Gold. This is a notable exception where

you are over 50% off.

Most of them it’s much simpler and in metals you tend to get very good

reaction energies.

MIT’s 3.320




N. Zabaras (3/5/2012) 35

COMPARING ENERGY OF STRUCTURES

For most elements, both LDA and GGA seem to predict the correct

structure for a material

Notable exceptions: Fe is FCC in LDA, not BCC. In GGA, that is correct.

Materials with substantial electron correlation effects (e.g. Pu)

High Throughput studies are now possible.

http://datamine.mit.edu/htap/alloys.jsp

If you go deep down in the periodic table, especially f-electron metals you start to

see failures of LDA and GGA. An important ion is Plutonium.





N. Zabaras (3/5/2012) 36

REDOX REACTIONS: Reaction of an oxide with a metal

The errors become bigger in redox reactions. The 3 reactions shown are

essentially a reaction between an oxide (phosphate) with a metal.

The reaction energies are not considerably off.

GGA gives you 2.8 eV for the first reaction, the experiment is 3.5. The third

reaction which is very similar gives an error of 30%. You get 3.3 eV, the

experiment is 4.6. Why is that?

MIT’s 3.320




N. Zabaras (3/5/2012) 37

REDOX REACTIONS

It has to do with a lack of error cancellation.

You need to look in detail what happens to the electronic structure in these

materials. These are redox reactions. If you do the math on the valences (lets

say for the first reaction), the iron on FePO4 is Fe3+, Phosphorous is P5+ and

Oxygen is O2-. Iron in LiFePO4 is Fe2+ and Lithium is Li1+. What has happened

in this reaction?

MIT’s 3.320




N. Zabaras (3/5/2012) 38

Transferring electrons from a delocalized to a localized state

All these reactions involve the transfer of an electron from a

delocalized state in Li metal to a localized state in the transition

metal oxide (phosphate)

You’ve taken essentially an electron from Lithium in its metallic state (and in the

process ionize Lithium) , and put it on the Iron Fe3+ to make Iron Fe2+.

That electron in Lithium (alkaline metal) is in a wide delocalized orbital and you

are transferring it to a localized d-state on the iron. That electron is essentially

being transferred between extremely different states. You essentially transfer

between such different states that you start losing the cancellation of errors that

you need in DFT.

In particular, the error here comes from what we call self-interaction error.



N. Zabaras (3/5/2012) 39

SELF INTERACTION ERROR

Remember that in DFT, we reduce the problem to 1-electron equations where you

have the kinetic energy , the nuclear potential and then the effective

potential Veff(ri)



N. Zabaras (3/5/2012) 40

Self-interaction effect in DFT

The effective potential has the exchange correlation in it but also has the Hartree

field i.e. the Coulombic field from all the electrons. That field includes the

electron itself.

When you calculate the charge density that’s the charge density of all the

electrons. When you calculate the potential coming from that charge density,

that’s the potential coming from all the electrons but you operate that now on a

single electron.



N. Zabaras (3/5/2012) 41

SELF INTERACTION ERROR IN DFT

The electron is feeling its own potential. Part of the exchange correlation term

corrects for that but not all of it. And the problem is that the correction does not

operate as well on different forms of the charge density.

In a metal you have a small self-interaction error and the reason is that a state

in a metal is sort of very delocalized -- has very spread out charge density. If

you want to think of it, part of the electron in one of the locations marked in the

figure below doesn’t feel much of the potential coming from the other marked

piece of the charge density because they are very far away.

X X



N. Zabaras (3/5/2012) 42

SELF INTERACTION ERROR IN DFT

However, if you consider a very localized state, the potential from the electron is

very high where the electron itself is sitting because it’s all very close. If you put

an electron in a Delta function, if you could, if you didn’t have an uncertainty

principle and you calculated its potential it’s basically sitting on top of itself.

Then you would have an infinite self-interaction.

So, the more local the state is, the more self-interaction you have and the

exchange correlation functional can’t quite correct these 2 in the same

way.



N. Zabaras (3/5/2012) 43


Remember that the exchange correlation correction comes from homogeneous

charge densities. So it tends to correct the metallic state better than the

localized state.

This is why the redox reactions showed a big error because we were

transferring from a state that was metallic, the electron went from the Lithium

state to the transitional metal state. Somehow the self-interaction error

doesn’t cancel.

You will see things like that whenever you transfer electrons between

quite different states.



N. Zabaras (3/5/2012) 44

LDA Vs. GGA: Summary of results for property calculation

Summary (LDA)

Lattice constants: 1-3% too small

Cohesive Energies: 5-20% too strongly bound

Bulk Modulus: 5-20% (largest errors for late TM)

Bandgaps: too small

GGA gives better cohesive energies. Effect on lattice

parameters is more random. GGA important for magnetic systems

When you work in metals and semi-conductors (which tend to be fairly

delocalized states) LDA and GGA do quite well for the lattice parameters and

even cohesive energies.

In more complicated materials, we have learned more about the errors of LDA

and GGA.

There is a series of methods under development to deal better with the

correlation energy and the self-interaction energy to solve these problems of

both energetics and also electronic structure.

spin polarized version of dft & accuracy studies of dft

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