Is a triplet state or singlet state more stable?

Is a triplet state or singlet state more stable? How can an atom have a long lifetime in a triplet state even though two electrons have the same spin? 




The key issue here that we are comparing excited states. So let's begin by defining Singlet and Triplet Excited States.
Singlet and Triplet Excited States: A singlet or a triplet can form when one electron is excited to a higher energy level. In an excited singlet state, the electron is promoted in the same spin orientation as it was in the ground state (paired). In a triplet excited stated, the electron that is promoted has the same spin orientation (parallel) to the other unpaired electron. The difference between the spins of ground singlet, excited singlet, and excited triplet is shown in the illustrative Figure below. Singlet, doublet and triplet is derived using the equation for multiplicity, $2S+1$, where $S$ is the total spin angular momentum (sum of all the electron spins). Individual spins are denoted as spin up ($s = +1/2$) or spin down ($s = -1/2$). If we were to calculated the S for the excited singlet state, the equation would be $2[(+1/2) + (-1/2)]+1 = 2(0)+1 = 1$, therefore making the center orbital in the figure a singlet state. If the spin multiplicity for the excited triplet state was calculated, we obtain $2[(+1/2) + (+1/2)]+1 = 2(1)+1 =3$, which gives a triplet state as expected.


![enter image description here](https://i.stack.imgur.com/PknVI.png)


Illustrative Figure: Spin in the ground and excited states The difference between a molecule in the ground and excited state is that the electrons is diamagnetic in the ground state and paramagnetic in the triplet state.This difference in spin state makes the transition from singlet to triplet (or triplet to singlet) more improbable than the singlet-to-singlet transitions. This singlet to triplet (or reverse) transition involves a change in electronic state. \*\*\*For this reason, the lifetime of the triplet state is longer than the singlet state by approximately 104 seconds fold difference.\*\*\*The radiation that induced the transition from ground to excited triplet state has a low probability of occurring, thus their absorption bands are less intense than singlet-singlet state absorption. The excited triplet state can be populated from the excited singlet state of certain molecules which results in phosphorescence. These spin multiplicities in ground and excited states can be used to explain transition in photoluminescence molecules by the Jablonski diagram.


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If you consider stability simply in terms of energy and *not* lifetime, then a triplet state is definitely more stable than a singlet state, i.e., a triplet state is of lower energy than a singlet excited state. 


This can be understood when we look at the energy contributions in a singlet and a triplet. Consider a simple case with two electrons labeled as $1$ and $2$ and two spin orbitals $a$ and $b$. In the singlet state electron $1$ occupies orbital $a$ with $1/2$ spin and electron $2$ occupies orbital $b$ with $-1/2$ spin. In the triplet state, $1$ occupies $a$ and $2$ occupies $b$, both with the same spin. The energy of the singlet state is given as:


$E\_S = h\_{11}+h\_{22}+J\_{12}$


And the energy of the triplet state is given as:


$E\_T = h\_{11}+h\_{22}+J\_{12}-K\_{12}$


Here $h\_{nn}$ is a contribution to energy irrespective of the spin of the electron. $J\_{nm}$ is the coulomb term which exists between every unique pair of electrons. $K\_{nm}$ is the exchange term, which only exists for an electron pair of the same spin. The exchange term has no easy classical interpretation. But importantly, it is a positive number, which implies that $-K\_{nm}$ always decreases the energy.


Therefore, $E\_S$ is greater than $E\_T$, meaning a triplet state is more stable compared to a singlet state.


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> 
> Is triplet state stable or singlet state ??
> 
> 
> 


It depends upon the spin system (atom, molecule, etc.).


Diamagnetic compounds have singlet ground states, so for many compounds the ground states are singlets.


Examples of triplet ground states include dioxygen. [Singlet dioxygen](http://en.wikipedia.org/wiki/Singlet_oxygen) is a excited state.


For dihydrogen, the singlet state is the ground state, but at room temperature, there is an equilibrium concentration of 75% triplet state and 25% ground state. Near zero kelvins, the singlet state dominates, but the energy difference between the states is so small that, in accordance with Boltzmann statistics, at room temperature the states are populated in proportion to the number of microstates, the triplet having 3 and the singlet having 1. But in this spin system, the spins are proton spins not electron spins.


[Methylene](http://en.wikipedia.org/wiki/Methylene_(compound)) also has a triplet ground electronic state and singlet excited state.



> 
> How can an atom stay long time in a triplet state though the electrons have same spin?? 
> 
> 
> 


Firstly, electronic "triplet state" doesn't exactly mean the electrons have the same spin. The three possibilities referred to by the word "triplet" are 1. both spins are up 2. both spins are down and 3. $1/\sqrt2$(up,down + down,up)


In any case, the lifetime of any excited state is not fixed even for a particular spin system. Mechanisms for the system to transition to the ground state need to be considered. In the gas phase the excited state may be long lived but it could be many orders of magnitude less in a liquid solvent. 


Just to give an example, for atomic helium in the triplet 1s2s state, the lifetime to decay by radiation to the 1s1s singlet state is 7859 seconds.


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Stability is about the energy level which is a thermodynamic factor while lifetime is a kinetic factor. That is why they are not necessarily agree with each other.
Triplet live longer simply because it is kinetically harder for the excited electron to jump back to the original orbital and at the same time flip its spinning state. It is not really connected to the stability of it. Stability is more complicated depending many other aspects.


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