atomic electronic states

8/6/2019 Atomic Electronic States

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Atomic Electronic States

By virtue of their motion about the nucleus, electrons possess orbital

angular momentum. The orbital angular momentum of the electron, l,

is a vector and is the vector cross product of the vectors; the linearmomentum, p, and the radius, r, of the orbital motion. When

explicitly discussing vectors, the symbols for vectors will be bolded.

The angular momentum is perpendicular to the plane defined by the

linear momentum and radius vectors and has a direction defined by

the right hand rule ( is the angle between the r and p vectors):

Electrons also possess spin angular momentum. While it helps to

think of the spin angular momentum as you would a spinning top,

this is purely a fiction, as the electron spin angular momentum, s,

which is also a vector, is a non-classical affect that arises naturally

in the relativistic treatment of the motion of electrons in atoms.

The electron spin angular momentum can be aligned in either a

positive or negative sense along an external or laboratory defined

axis (usually designated as the z axis) giving rise to the electron spin

quantum numbers:

ms = + a spin up electronms = - a spin down electron

pr

l = r x p = r p sin


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neutrons, also with spin 1/2, are other examples of fermions.

Particles with integral spin are

symmetric with respect to the exchange

of these particles in a wavefunction and

are known asbosons (after Satyendra

Bose pictured on the stamp at right).

Photons with spin 1 and deuterons with

spin 0 are examples of bosons. There

is no restriction on the number ofbosons that can occupy a given state.

Particles with half integral spin are

anti-symmetric with respect to the

exchange of these particles in a

wavefunction and are known as

fermions (after Enrico Fermi

pictured on the left). Electrons have

an intrinsic spin of 1/2 and are

therefore fermions. Protons and


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Because they are anti-symmetric with respect to exchange no two

fermions can occupy the same state. This restriction on fermions leads

to the Pauli Exclusion Principle (after Wolfgang Pauli shown below):

Or put another way no two electrons can have the same four quantumnumbers and hence only two electrons; one spin up with ms = +1/2

and the other spin down with ms = - 1/2, can occupy a spatial orbital,

e.g., a 1s atomic orbital, as defined by the quantum numbers; n, l, and

ml:

1s atomic orbital (n=0, l=0, ml=0)

All electronic wave functions must

be anti-symmetric with respect to

the exchange of any two electrons.


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For a given electronic configuration the electrons orbital and spin

angular momentum give rise to sets of microstates associated with

that electronic configuration that have different energies.

Consider a given electronic configuration, e.g., for neutral carbon, C I(the I indicates that the C atom is unionized; a II would imply a

singly ionized carbon atom or C+):

neutral C or C I: 1s2 2s2 2p2 or [He] 2s2 2p2

Here [He] is a shorthand for 1s2 , i.e., the completed 1stperiod in theperiodic table.

A micro state is any arrangement of electrons consistent with a given

electronic configuration that does not violate the Pauli Exclusion

Principle.

Two microstates associated with this electronic configuration that do

not violate the Pauli Exclusion Principle and have different electronic

energies are:

2p2 __ __

2p2

__and

2s2 2s2

1s2 1s2

Based on this diagram is energy required to pair electrons or is it theother way around? Give a physical explanation for your answer.

E


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In order to identify the particular sets of microstates that give rise to

the possible electronic energies associated with a given electronic

configuration we need to develop the set of scalar values of the

electrons total orbital angular momentum, L, and total spin angular

momentum, S, that account for all the microstates.

First consider the contribution to L and S that come from electrons

that are paired in filled shells and sub-shells, e.g.:

[He] 2s2 or 1s2 2s2

Since each electron is in an s atomic orbital, where the orbital angular

momentum, l, is equal to zero; and hence the magnetic quantum

number, ml, is also equal to zero, each electron has:

l= 0 and ml = 0 | ml | | l |

The sum of these individual mlgives a value for ML, the projection

of the total orbital angular momentum vector, L, on an externally

defined z axis:

ML = ml, i = 0

Since this is the only possible value of ML for any set of filled shellor sub-shell configurations, it is also the maximum value, which is

equal to the scalar value of L, hence:

L = 0

for any set of filled shell or sub-shell electronic configurations.


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A similar argument applies to the development of the scalar value

of the total spin angular momentum, S, for any set of filled shells or

sub-shells.

Since each electron in a filled atomic orbital with two electrons will

have one electron with a spin quantum number of + and the

other of , their sum will zero, e.g.:

1s2 ms, i = (+1/2) + (-1/2) = 0

Thus for any set of filled shells or sub-shells the value MS, the

projection of the total spin angular momentum vector, S, on an

externally defined z axis:

MS = ms, i = 0

Since this is the only possible value of MS for any set of filled shell

or sub-shell configurations, it is also the maximum value, which is

equal to the scalar value of S, hence:

S = 0

for any set of filled shell or sub-shell electronic configurations.

Any set of filled shells of sub-shells always contributes zero to the

calculation of the scalar values of L and S and therefore all filled

shells and sub-shells can be ignored in determining L and S for

electronic configurations with a partially filled sub-shell!


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Thus in determining the scalar values of L and S for any electronic

configuration involving partially filled sub-shells we only need to

focus on the partially filled subshells, e.g., for neutral carbon, C I, we

only need to consider microstates like:

2p2 __

Since there are three possible magnetic quantum numbers for a p

atomic orbital:

__ __ __

ml = +1 0 -1

and each electron can be either spin up, , or spin down, , in these

orbitals, there are 6 possible states for each electron and since in a p2

electronic configuration there are two electrons that can be placed in

these 6 states there are:

6 ! / [ 2 ! ( 6 2 ) ! ] = 15 microstates associated with a

p2 electronic configuration


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To calculate the electronic states associated with C I we first

enumerate in some systematic way the 15 possible microstates

associated with the partially filled 2p2 subshell:

ml = +1 0 -1 ML MS

1 2 0

2 1 1

3 1 0

4 0 1

5 0 06 1 0

7 1 -1

8 0 0

9 0 -1

10 0 0

11 -1 112 -1 0

13 -1 0

14 -1 -1

15 -2 0

and calculate the:

ML = ml, i

MS = ms, I

associated with each microstate.


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Now pick out the maximum ML from these set of micro states?

ML, max = +2

Since the ML are the projections of the electrons total orbital

angular momentum vector, L, on the externally defined z axis:

ML = -2

ML = -1

ML = 0

ML = +1

ML = +2

L

z

It should be clear that the scalar value of L is equal to ML, max:

L = ML, max = +2

Electronic states associated with particular scalar values of the

scalar value of the total orbital angular momentum , L ,are given

letter designations:

L = 0 1 2 3 4 5 6 7

S P D F G H I K

Thus we know that an L = 2 or D state exists for C I.


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Now look for the maximum MS associated with ML = +2:

MS, max = 0 (for ML = +2)

Since, like ML and L, the MS are the projections of the electronstotal spin angular momentum vector, S, on the externally defined z

axis, an:

MS, max = 0 => S = 0

The value of the total spin angular momentum vector, S, is usually

encoded as the spin multiplicity:

spin multiplicity = 2 S + 1 = 2 (0) + 1 = 1 (a singlet state)

which is equal to the number of values that MS can take on where

MS can range from +S to -S, changing by integer values:

MS = +S, +S-1, , +1, 0, -1, -S+1, -S


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The scheme for determining the electronic states that we have

described so far is known as Russell-Saunders coupling and is

applicable to the lighter elements.

In Russell-Saunders coupling orLS the orbital angular momentum

vectors of each electron, li, couple to form the total angular

momentum vector, L; and the spin angular momentum vector of

each electron, si, also couple to form the total spin angular

momentum vector, S. These L and S vectors in turn couple to form

the total angular momentum vector, J (not to be confused with the

rotational quantum number).

The allowed values of J are given by:

J = |L+S|, |L+S|-1, , +1, 0, -1, |L-S|+1, |L-S|

For the singlet D state, that we have identified as one of theelectronic states associated with the ground electronic configuration

of C I, [He] 2s2 2p2, we have the following possible values of J:

J = |2+0| = 2 to |2-0| = 2

or a single value of J = 2.

When either L or S is 0, how many values of J will there be?


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For heavy atoms angular momenta couple according to JJ coupling.

In JJ coupling the individual electron orbital, li, and spin, si, angular

momenta vectors first couple to give a individual electron total

angular momentum,ji:

ji = li + si,

which in turn add vectorially to give total angular momentum for the

atomic species:

J = j i


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An electronic term symbol describing an atomic electronic state has

the general form:2S+1LJ

where 2S+1 is the spin multiplicity and L is the appropriate letter

designation for the total orbital angular momentum.

The electronic term symbol describing the particular electronic

state associated with the ground electronic configuration of C I,

[He] 2s2 2p2 that we have just developed is a singlet D two state:

1D2

This 1D state (for all possible values of J for this state) accounts for:

(2S+1) (2L+1) = (2(0)+1) (2(2)+1) = 5

of the 15 microstates associated with C I, [He] 2s2 2p2 (notice that

the 1st term in the above calculation is just the spin multiplicity).


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Now associate the 1D term (without regard to J) with 5 of the

microstates consistent with this term (these states must range from

ML = +2 to ML = -2 by integer values and must all have MS = 0):

ml = +1 0 -1 ML MS

1 2 0 1D

2 1 1

3 1 0 1D

4 0 1

5 0 0 1D6 1 0

7 1 -1

8 0 0

9 0 -1

10 0 0

11 -1 112 -1 0 1D

13 -1 0

14 -1 -1

15 -2 0 1D

There may be multiple valid choice for1D and it doesnt matterwhich valid choice you make, as long as you assign exactly 5

microstates to 1D.


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To determine the term symbols associated with the remaining

electronic states of C I, [He] 2s2 2p2 we repeat this process

(beginning from page 9 of this presentation) for any unassigned

microstates continuing in this fashion until all the microstates have

been assigned.

What is the largest unassigned value of ML?

What scalar value of L does this imply must be present?

What is the largest MS associated with the ML you have just

identified?

What scalar value of S is associated with this MS

?

What is the spin multiplicity associated with this state?

What are the allowed values of J for this state?

What are the electronic term symbols in this case?

How many microstates of C I, [He] 2s2 2p2 does this electronic state

account for?


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Now associate the 3P term with 9 of the microstates consistent with

this term (these states must range from ML = +1 to ML = -1 by integer

values and each of these ML must have an MS that ranges from

MS = +1 to MS = -1, also by integer values):

ml = +1 0 -1 ML MS

1 2 0 1D

2 1 1 3P

3 1 0 1D

4 0 1 3P5 0 0 1D

6 1 0 3P

7 1 -1 3P

8 0 0 3P

9 0 -1 3P

10 0 011 -1 1 3P

12 -1 0 1D

13 -1 0 3P

14 -1 -1 3P

15 -2 0 1D

What is the final electronic term symbol associated with the single

remaining unassigned microstate of C I, [He] 2s2 2p2 ?


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The complete set of electronic states, each with a different energy,

associated with ground state electronic configuration of neutral

carbon, C I, [He] 2s2 2p2 , are:

1D2 , 3P2 , 3P1 , 3P0 , 1S0

The relative ordering of these states can be

determined by applying Hunds rules (see

photo at left), which are applicable where

Russell-Saunders coupling is valid.

Rule 1: Electronic states with the largest total

spin angular momentum, S, lie lowest in

energy:

1D2 ,1S0

3P2 ,3P1 ,

3P0E

Rule 2: For electronic states with the same total spin angular

momentum, S, states with the largest total orbital angular

momentum,L, lie lowest in energy:

1S0

1D2

3P2 ,3P1 ,

3P0

E


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momentum, S, and the same total orbital angular momentum,L,

* that are less than half filled, states with the smallest total

angular momentum, J, lie lowest in energy;

* that are more than half filled, states with the largest total

angular momentum, J, lie lowest in energy.

Complete the relative ordering in energy of electronic states, each

with a different energy, associated with ground state electronicconfiguration of neutral carbon, C I, [He] 2s2 2p2 :

1S0

1D2

The procedures that have illustrated for determining the electronic

states associated with a given electronic configuration apply not onlyto neutral atoms in their ground state, but also to atomic exited states

and atomic ions.

For atomic systems with greater than half-filled shells, the holes in

these shells, i.e., the absence of electrons can be treated just they

were the only electrons present. For example, except for HundsRule #3, the results for O I, [He] 2s2 2p4 are the same as

C I, [He] 2s2 2p2 .

E


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The actual energy differences are measured relative to the electronic

ground state and are generally determined by experiment (and

sometimes by calculation):

Which state of C I, [He] 2s2 2p2 is the ground electronic state?

These experimental differences in energy between atomic electronic

states are tabulated for both ground and excited state electronic

configurations of neutral atoms and ions in which is accessible

through the NIST Chemistry Webbookunder:

Data at other public NIST sites:

NIST Atomic Spectra Database - Lines Holdings (on

physics web site)

Compare the relative ordering of the electronic states of that you

determined for C I, [He] 2s2 2p2 with that found in the NISTChemistry Webbook


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Grotrian diagrams are a way of visualizing the relative energies and

allowed states associated with a particular element and the allowed

or observed electronic transitions between those states.

In a Grotrian diagram energy is plotted on the vertical axis and theelectronic states are plotted as short horizontal lines at the

appropriate energy and are grouped and ordered by listing states

with highest spin multiplicity on the left, followed by states with

with lowest total orbital angular momentum, L, and finally by

states with highest total angular momentum, J.

The Grotrian diagram for the electronic states ofNa I is:

The NIST Chemistry Webbookunder the NIST Atomic Spectradatabase and further under the link Lines has the ability to create

Grotrian diagrams.

Th i i t iti h i b ld d b l f th it d


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Na I [Ne] 3p1

Na I [Ne] 3s1

The emission transitions shown in bold red below from the excited

electronic configuration [Ne] 3p1 to an ground state electronic

configuration [Ne] 3s1 ofNa I:

2P3/2

(Na I [Ne] 3p1) 2S1/2

(Na I [Ne] 3s1)

2P1/2 (Na I [Ne] 3p1) 2S1/2 (Na I [Ne] 3s

1)

are the famous Na doublet, responsible for the dominant yellow

color seen in flames where sodium is present.

The image at right, showing this sodium emission

doublet from an electric pickle resolved by a

spectrograph, was obtained by Benoit Minster of

Grenoble, France. Although it is the Na+ ion or

Na II that is present in the pickle, in the applied

electric field electronic configurations associatedwith neutral Na I are also present and are

responsible for the spectrum.

Wh t l t i t b l i i t d ith d l t i


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What electronic term symbol is associated with ground electronic

configuration of neutral Zn I, [Ar] 4s2 3d10?

For the ground electronic state of N I, [He] 2s2 2p3:

a. determine the term symbols of all electronic states

associated with this electronic configuration;

b. order these electronic states in terms of their relative

energy;

c. Compare your relative ordering with that found on the

NIST Atomic Spectra Database determine the term

symbols of all electronic states associated with this

electronic configuration.

For the excited electronic state of Mg I, [Ne] 3s1 3p1:

a. determine the term symbols of all electronic states

associated with this electronic configuration (note the 3s1

and 3p1 electrons must remain in their respective atomic

orbitals);

b. order these electronic states in terms of their relative

energy;

c. Compare your relative ordering with that found on the

NIST Atomic Spectra Database determine the termsymbols of all electronic states associated with this

electronic configuration.


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For the ground electronic configuration of the ion Cl+;

Cl II, [Ne] 3s2 3p4:

a. determine the term symbols of all electronic statesassociated with this electronic configuration;

b. order these electronic states in terms of their relative energy;

c. compare your relative ordering with that found on the NIST

Atomic Spectra Database determine the term symbols of all

electronic states associated with this electronic

configuration.

For the ground electronic state configuration ofN

b I, [Kr] 5s1

4d4

:

a. calculate the number of microstates;

b. determine the term symbol of the ground electronic state.

Take Aways


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Take Aways

The orbital angular momentum of the electron, l, is a vector and is

the vector cross product of the vectors; the linear momentum, p, and

the radius, r, of the orbital motion.

l = r x p

the electron spin angular momentum, s, which is also a vector, is a

non-classical affect that arises naturally in the relativistic treatment of

the motion of electrons in atoms that results in two electron spin

quantum numbers:

ms = + a spin up electron

ms = - a spin down electron

Bosons are particles with integral spin which are symmetric with

respect to the exchange of the particles.

Fermions are particles with half-integral spin which are anti-

symmetric with respect to the exchange of the particles.

The Pauli Exclusion Principle states that :

All electronic wave functions must be anti-symmetric with

respect to the exchange of any two electrons.

and dictates that only two electrons; one spin up with ms = +1/2 and

the other spin down with ms = - 1/2, can occupy a spatial orbital.


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For a given electronic configuration the electrons orbital and spin

angular momentum give rise to sets of microstates associated with

that electronic configuration that have different energies.

A micro state is any arrangement of electrons consistent with a given

electronic configuration that does not violate the Pauli Exclusion

Principle.

Any set of filled shells of sub-shells always contributes zero to thecalculation of the scalar values of L and S and therefore all filled

shells and sub-shells can be ignored in determining L and S for

electronic configurations with a partially filled sub-shell!

In Russell-Saunders coupling, which is applicable to the lighterelements, the orbital angular momentum vectors of each electron, li,

couple to form the total angular momentum vector, L; and the spin

angular momentum vector of each electron, si, also couple to form

the total spin angular momentum vector, S. These L and S vectors in

turn couple to form the total angular momentum vector, J.

Electronic states associated with particular scalar values of the

scalar value of the total orbital angular momentum , L ,are given

letter designations:

L = 0 1 2 3 4 5 6 7S P D F G H I K



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the general form:2S+1LJ

where 2S+1 is the spin multiplicity and L is the appropriate letterdesignation for the total orbital angular momentum.

The relative ordering of electronic states under Russell-Saunders

coupling) can be determined by applying Hunds rules:

Rule 1: Electronic states with the largest total spin angular

momentum, S, lie lowest in energy.


momentum, S, states with the largest total orbital angular

momentum,L, lie lowest in energy.


momentum, S, and the same total orbital angular momentum,L,

* that are less than half filled, states with the smallest

total angular momentum, J, lie lowest in energy;

* that are more than half filled, states with the largest

total angular momentum, J, lie lowest in energy.

For atomic systems with greater than half-filled shells, the holes in

these shells, i.e., the absence of electrons can be treated just theywere the only electrons present.

atomic electronic states

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