Fuchs's theorem: Difference between revisions

Latest revision as of 08:43, 19 March 2013

Hooke's atom, also known as harmonium or hookium, refers to an artificial helium-like atom where the Coulombic electron-nucleus interaction potential is replaced by a harmonic potential.^[1]^[2] This system is of significance as it is, for certain values of the force constant defining the harmonic containment, an exactly solvable^[3] ground-state many-electron problem that explicitly includes electron correlation. As such it can provide insight into quantum correlation (albeit in the presence of a non-physical nuclear potential) and can act as a test system for judging the accuracy of approximate quantum chemical methods for solving the Schrödinger equation.^[4]^[5] The name "Hooke's atom" arises because the harmonic potential used to describe the electron-nucleus interaction is a consequence of Hooke's law.

Definition

Employing atomic units, the Hamiltonian defining the Hooke's atom is

\hat{H} = - \frac{1}{2} \nabla_{1}^{2} - \frac{1}{2} \nabla_{2}^{2} + \frac{1}{2} k (r_{1}^{2} + r_{2}^{2}) + \frac{1}{| r_{1} - r_{2} |} .

As written, the first two terms are the kinetic energy operators of the two electrons, the third term is the harmonic electron-nucleus potential, and the final term the electron-electron interaction potential. The non-relativistic Hamiltonian of the helium atom differs only in the replacement:

- \frac{2}{r} \to \frac{1}{2} k r^{2} .

Solution

The equation to be solved is the two electron Schrödinger equation:

\hat{H} Ψ (r_{1}, r_{2}) = E Ψ (r_{1}, r_{2}) .

For arbitrary values of the force constant, k, the Schrödinger equation does not have an analytic solution. However, for a countably infinite number of values, such as k=¼, simple closed form solutions can be derived.^[5] Given the artificial nature of the system this restriction does not hinder the usefulness of the solution.

To solve, the system is first transformed form the Cartesian electronic coordinates, (r₁,r₂), to the center of mass coordinates, (R,u), defined as

R = \frac{1}{2} (r_{1} + r_{2}), u = r_{2} - r_{1} .

Under this transformation, the Hamiltonian becomes separable – that is, the |r₁ - r₂| term coupling the two electrons is removed (and not replaced by some other form) allowing the general separation of variables technique to be applied to further a solution for the wave function in the form $Ψ (r_{1}, r_{2}) = χ (R) Φ (u)$ . The original Schrödinger equation is then replaced by:

(- \frac{1}{4} \nabla_{R}^{2} + k R^{2}) χ (R) = E_{R} χ (R),

(- \nabla_{u}^{2} + \frac{1}{4} k u^{2} + \frac{1}{u}) Φ (u) = E_{u} Φ (u) .

The first equation for $χ (R)$ is the Schrödinger equation for an isotropic quantum harmonic oscillator with ground-state energy $E_{R} = (3 / 2) \sqrt{k} E_{h}$ and (unnormalized) wave function

χ (R) = e^{- \sqrt{k} R^{2}} .

Asymptotically, the second equation again behaves as a harmonic oscillator of the form $\exp (- (\sqrt{k} / 4) u^{2})$ and the rotationally invariant ground state can be expressed, in general, as $Φ (u) = f (u) \exp (- (\sqrt{k} / 4) u^{2})$ for some function $f (u)$ . It was long noted that f(u) is very well approximated by a linear function in u.^[2] Thirty years after the proposal of the model an exact solution was discovered for k=¼,^[3] and it was seen that f(u)=1+u/2. It was latter shown that there are many values of k which lead to an exact solution for the ground state,^[5] as will be shown in the following.

Decomposing $Φ (u) = R_{l} (u) Y_{l m}$ and expressing the Laplacian in spherical coordinates,

(- \frac{1}{u^{2}} \frac{\partial}{\partial u} (u^{2} \frac{\partial}{\partial u}) + \frac{{\hat{L}}^{2}}{u^{2}} + \frac{1}{4} k u^{2} + \frac{1}{u}) R_{l} (u) Y_{l m} (\hat{u}) = E_{l} R_{l} (u) Y_{l m} (\hat{u}),

one further decomposes the radial wave function as $R_{l} (u) = S_{l} (u) / u$ which removes the first derivative to yield

- \frac{\partial^{2} S_{l} (u)}{\partial u^{2}} + (\frac{l (l + 1)}{u^{2}} + \frac{1}{4} k u^{2} + \frac{1}{u}) S_{l} (u) = E_{l} S_{l} (u) .

The asymptotic behavior $S_{l} (u) \sim e^{- \frac{\sqrt{k}}{4} u^{2}}$ encourages a solution of the form

S_{l} (u) = e^{- \frac{\sqrt{k}}{4} u^{2}} T_{l} (u) .

The differential equation satisfied by $T_{l} (u)$ is

- \frac{\partial^{2} T_{l} (u)}{\partial u^{2}} + \sqrt{k} u \frac{\partial T_{l} (u)}{\partial u} + (\frac{l (l + 1)}{u^{2}} + \frac{1}{u} + (\frac{\sqrt{k}}{2} - E_{l})) T_{l} (u) = 0 .

This equation lends itself to a solution by way of the Frobenius method. That is, $T_{l} (u)$ is expressed as

T_{l} (u) = u^{m} \sum_{k = 0}^{\infty} a_{k} u^{k} .

for some $m$ and ${a_{k}}_{k = 0}^{k = \infty}$ which satisfy:

m (m - 1) = l (l + 1),

a_{0} \neq 0

a_{1} = \frac{a_{0}}{2 (l + 1)},

a_{2} = \frac{a_{1} + (\sqrt{k} (l + \frac{3}{2}) - E_{l}) a_{0}}{2 (2 l + 3)} = \frac{a_{0}}{2 (2 l + 3)} (\frac{1}{2 (l + 1)} + \sqrt{k} (l + \frac{3}{2}) - E_{l}),

a_{3} = \frac{a_{2} + (\sqrt{k} (l + \frac{5}{2}) - E_{l}) a_{1}}{6 (l + 2)},

a_{n + 1} = \frac{a_{n} + (\sqrt{k} (l + \frac{1}{2} + n) - E_{l}) a_{n - 1}}{(n + 1) (2 l + 2 + n)} .

The two solutions to the indicial equation are $m = l + 1$ and $m = - l$ of which the former is taken as it yields the regular (bounded, normalizable) wave function. For a simple solution to exist, the infinite series is sought to terminate and it is here where particular values of k are exploited for an exact closed-form solution. Terminating the polynomial at any particular order can be accomplished with different values of k defining the Hamiltonian. As such there exists an infinite number of systems, differing only in the strength of the harmonic containment, with exact ground-state solutions. Most simply, to impose a_k = 0 for k ≥ 2, two conditions must be satisfied:

\frac{1}{2 (l + 1)} + \sqrt{k} (l + \frac{3}{2}) - E_{l} = 0,

\sqrt{k} (l + \frac{5}{2}) = E_{l} .

These directly force a₂ = 0 and a₃ = 0 respectively, and as a consequence of the three term recession, all higher coefficients also vanish. Solving for $\sqrt{k}$ and $E_{l}$ yields

\sqrt{k} = \frac{1}{2 (l + 1)},

E_{l} = \frac{2 l + 5}{4 (l + 1)},

and the radial wave function

T_{l} = u^{l + 1} (a_{0} + \frac{a_{0}}{2 (l + 1)} u) .

Transforming back to $R_{l} (u)$

R_{l} (u) = \frac{T_{l} (u) e^{- \frac{\sqrt{k}}{4} u^{2}}}{u} = u^{l} (1 + \frac{1}{2 (l + 1)} u) e^{- \frac{\sqrt{k}}{4} u^{2}},

the ground-state (with $l = 0$ and energy $5 / 4 E_{h}$ ) is finally

Φ (u) = (1 + \frac{u}{2}) e^{- u^{2} / 4} .

Combining, normalizing, and transforming back to the original coordinates yields the ground state wave function:

Ψ (r_{1}, r_{2}) = \frac{1}{2 \sqrt{8 π^{5 / 2} + 5 π^{3}}} (1 + \frac{1}{2} | r_{1} - r_{2} |) \exp (- \frac{1}{4} (r_{1}^{2} + r_{2}^{2})) .

The corresponding ground-state total energy is then $E = E_{R} + E_{u} = \frac{3}{4} + \frac{5}{4} = 2 E_{h}$ .

Remarks

The exact ground state electronic density of the Hooke atom is^[4]

ρ (r) = \frac{2 π^{3 / 2}}{(8 + 5 \sqrt{π})} e^{- (1 / 2) r^{2}} ({(\frac{π}{2})}^{1 / 2} (\frac{7}{4} + \frac{1}{4} r^{2} + (r + \frac{1}{r}) e r f (\frac{r}{\sqrt{2}})) + e^{- (1 / 2) r^{2}}) .

From this we see that the radial derivative of the density vanishes at the nucleus. This is in stark contrast to the real (non-relativistic) helium atom where the density displays a cusp at the nucleus as a result of the unbounded Coulomb potential.

References

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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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↑ ^2.0 ^2.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ ^3.0 ^3.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ ^4.0 ^4.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ ^5.0 ^5.1 ^5.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[1] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[kestner62-2] 2.0 ^2.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[kais89-3] 3.0 ^3.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[kais93-4] 4.0 ^4.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[taut93-5] 5.0 ^5.1 ^5.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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-Bonus:  WP Twin and WP Twin Auto Backup: (link to  )  While not a theme, I think this software is essential if you are maintaining your Wordpress blog or regularly create new blog sites. You can either install Word - Press yourself or use free services offered on the web today. The effect is to promote older posts by moving them back onto the front page and into the rss feed. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. If you are happy with your new look then click "Activate 'New Theme'" in the top right corner. <br><br>These folders as well as files have to copied and the saved. The higher your blog ranks on search engines, the more likely people will find your online marketing site.  If you adored this article and you would such as to obtain more facts pertaining to [http://d2.ae/backup_plugin_82204 backup plugin] kindly check out our webpage. We also help to integrate various plug-ins to expand the functionalities of the web application. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. Many times the camera is following Mia, taking in her point of view in almost every frame. <br><br>It is also popular because willing surrogates,as well as egg and sperm donors,are plentiful. s cutthroat competition prevailing in the online space won. For a much deeper understanding of simple wordpress themes", check out  Upon browsing such, you'll be able to know valuable facts. Thousands of plugins are available in Word - Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. If you have any questions on starting a Word - Press food blog or any blog for that matter, please post them below and I will try to answer them. <br><br>If all else fails, please leave a comment on this post with the issue(s) you're having and help will be on the way. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. See a product, take a picture, and it gives you an Amazon price for that product, or related products. <br><br>You will know which of your Word - Press blog posts are attracting more unique visitors which in turn will help you develop better products and services for your customers. Here's a list of some exciting Word - Press features that have created waves in the web development industry:. Word - Press can also be quickly extended however improvement API is not as potent as Joomla's. with posts or testimonials updated as they are uploaded to a particular section of the website. However, if you're just starting out your blog site or business site, you can still search for an ideal theme for it without breaking your bank account.
+'''Hooke's atom''', also known as '''harmonium''' or '''hookium''', refers to an artificial [[helium]]-like atom where the [[Coulomb's law|Coulombic]] electron-nucleus interaction potential is
+replaced by a [[Quantum harmonic oscillator|harmonic potential]].<ref>{{cite book |last=Lucjan |first=Piela  |title=Ideas of Quantum Chemistry |publisher=[[Elsevier]] |location=Amsterdam  |isbn=978-0-444-52227-6 |pages=185–188 |year=2007 }}</ref><ref name="kestner62">{{cite journal |title = Study of Electron Correlation in Helium-Like Systems Using an Exactly Soluble Model |year = 1962 |author = N. R. Kestner and O. Sinanoglu |journal = Phys. Rev.|volume = 128|pages = 2687–2692 |doi = 10.1103/PhysRev.128.2687 |issue = 6|bibcode = 1962PhRv..128.2687K }}</ref> This system is of significance as it is, for certain values of the force constant defining the harmonic containment, an exactly solvable<ref name="kais89">{{cite journal |title = Dimensional scaling as a symmetry operation |year = 1989 |author = S. Kais, D. R. Herschbach, and R. D. Levine |journal = J. Chem. Phys.|volume = 91|pages = 7791 |doi = 10.1063/1.457247 |issue = 12|bibcode = 1989JChPh..91.7791K }}</ref> ground-state [[Many-body problem|many-electron problem]] that explicitly includes [[Electronic correlation|electron correlation]]. As such it can provide insight into quantum correlation (albeit in the presence of a non-physical nuclear potential) and can act as a test system for judging the accuracy of [[Computational chemistry|approximate quantum chemical methods]] for solving the [[Schrödinger equation]].<ref name="kais93">{{cite journal |title = Density functionals and dimensional renormalization for an exactly solvable model |year = 1993 |author = S. Kais, D. R. Herschbach, N. C. Handy, C. W. Murray, and G. J. Laming |journal = J. Chem. Phys. |volume = 99 |pages = 417 |doi = 10.1063/1.465765|bibcode = 1993JChPh..99..417K }}</ref><ref name="taut93">{{cite journal |title = Density functionals and dimensional renormalization for an exactly solvable model |year = 1993 |author = M. Taut |journal = Phys. Rev. A |volume = 48|pages = 3561–3566 |doi = 10.1103/PhysRevA.48.3561 |pmid = 9910020 |issue = 5|bibcode = 1993PhRvA..48.3561T }}</ref>  The name "Hooke's atom" arises because the harmonic potential used to describe the electron-nucleus interaction is a consequence of [[Hooke's law]].
+==Definition==
+Employing [[atomic units]], the [[Molecular Hamiltonian|Hamiltonian]] defining the Hooke's atom is
+:<math>\hat{H} = -\frac{1}{2}\nabla^{2}_{1} -\frac{1}{2}\nabla^{2}_{2} + \frac{1}{2}k(r^{2}_{1}+r^{2}_{2}) + \frac{1}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}.</math>
+As written, the first two terms are the kinetic energy operators of the two electrons, the third term is the harmonic electron-nucleus potential, and the final term the electron-electron interaction potential. The non-relativistic Hamiltonian of the helium atom differs only in the replacement:
+:<math>-\frac{2}{r} \rightarrow \frac{1}{2}kr^{2}.</math>
+==Solution==
+The equation to be solved is the two electron Schrödinger equation:
+:<math>\hat{H}\Psi(\mathbf{r}_{1},\mathbf{r}_{2}) = E\Psi(\mathbf{r}_{1},\mathbf{r}_{2}).</math>
+For arbitrary values of the force constant, ''k'', the Schrödinger equation does not have an analytic solution. However, for a [[Countable set|countably infinite]] number of values, such as ''k''=¼, simple closed form solutions can be derived.<ref name="taut93"/> Given the artificial nature of the system this restriction does not hinder the usefulness of the solution.
+To solve, the system is first transformed form the Cartesian electronic coordinates, ('''r'''<sub>1</sub>,'''r'''<sub>2</sub>), to the center of mass coordinates, ('''R''','''u'''), defined as
+:<math>\mathbf{R}=\frac{1}{2}(\mathbf{r}_{1}+\mathbf{r}_{2}), \mathbf{u}=\mathbf{r}_{2}-\mathbf{r}_{1}.</math>
+Under this transformation, the Hamiltonian becomes separable – that is, the |'''r'''<sub>1</sub> - '''r'''<sub>2</sub>| term coupling the two electrons is removed (and not replaced by some other form) allowing the general [[separation of variables]] technique to be applied to further a solution for the wave function in the form <math>\Psi(\mathbf{r}_{1},\mathbf{r}_{2})=\chi(\mathbf{R})\Phi(\mathbf{u})</math>. The original Schrödinger equation is then replaced by:
+:<math>\left( -\frac{1}{4}\nabla^{2}_{\mathbf{R}}+kR^{2} \right)\chi(\mathbf{R}) = E_{\mathbf{R}}\chi(\mathbf{R}),</math>
+:<math>\left( -\nabla^{2}_{\mathbf{u}}+\frac{1}{4}ku^{2} +\frac{1}{u}\right)\Phi(\mathbf{u}) = E_{\mathbf{u}}\Phi(\mathbf{u}).</math>
+The first equation for <math>\chi(\mathbf{R})</math> is the Schrödinger equation for an isotropic [[quantum harmonic oscillator#Example:_3D_isotropic_harmonic_oscillator|quantum harmonic oscillator]] with ground-state energy <math>E_{\mathbf{R}}=(3/2)\sqrt{k} E_{\mathrm{h}}</math> and (unnormalized) wave function
+:<math>\chi(\mathbf{R}) = e^{-\sqrt{k}R^{2}}.</math>
+Asymptotically, the second equation again behaves as a harmonic oscillator of the form <math>\exp(-(\sqrt{k}/4)u^{2})\,</math> and the rotationally invariant ground state can be expressed, in general, as <math>\Phi(\mathbf{u})=f(u)\exp(-(\sqrt{k}/4)u^{2})\,</math> for some function <math>f(u)\,</math>. It was long noted that ''f''(''u'') is very well approximated by a linear function in ''u''.<ref name="kestner62"/> Thirty years after the proposal of the model an exact solution was discovered for ''k''=¼,<ref name="kais89"/> and it was seen that ''f''(''u'')=1+''u''/2. It was latter shown that there are many values of ''k'' which lead to an exact solution for the ground state,<ref name="taut93"/> as will be shown in the following.
+Decomposing <math>\Phi(\mathbf{u})=R_{l}(u)Y_{lm}</math> and expressing the [[Laplacian]] in [[spherical coordinates]],
+:<math>\left( -\frac{1}{u^{2}}\frac{\partial}{\partial u}\left(u^{2}\frac{\partial}{\partial u}\right) + \frac{\hat{L}^{2}}{u^{2}}   +\frac{1}{4}ku^{2} +\frac{1}{u}\right)R_{l}(u)Y_{lm}(\hat{\mathbf{u}}) = E_{l}R_{l}(u)Y_{lm}(\hat{\mathbf{u}}),</math>
+one further decomposes the radial wave function as <math>R_{l}(u)=S_{l}(u)/u\,</math> which removes the first derivative to yield
+:<math>-\frac{\partial^{2}S_{l}(u)}{\partial u^{2}}+\left(\frac{l(l+1)}{u^{2}}+\frac{1}{4}ku^{2}+\frac{1}{u}\right)S_{l}(u) = E_{l}S_{l}(u).</math>
+The asymptotic behavior <math>S_{l}(u) \sim e^{-\frac{\sqrt{k}}{4}u^{2}}\,</math> encourages a solution of the form
+:<math>S_{l}(u) = e^{-\frac{\sqrt{k}}{4}u^{2}}T_{l}(u).</math>
+The differential equation satisfied by <math>T_{l}(u)\,</math> is
+:<math>-\frac{\partial^{2}T_{l}(u)}{\partial u^{2}} + \sqrt{k}u\frac{\partial T_{l}(u)}{\partial u} + \left(\frac{l(l+1)}{u^{2}}+\frac{1}{u}+\left(\frac{\sqrt{k}}{2}-E_{l}\right)\right)T_{l}(u) = 0.</math>
+This equation lends itself to a solution by way of the [[Frobenius method]]. That is,  <math>T_{l}(u)\,</math> is expressed as
+:<math>T_{l}(u) = u^{m}\sum_{k=0}^{\infty}\ a_{k}u^{k}.</math>
+for some <math>m\,</math> and <math>\{a_{k}\}_{k=0}^{k=\infty}\,</math> which satisfy:
+:<math>m(m-1) = l(l+1)\,, </math>
+:<math>a_{0} \neq 0\, </math>
+:<math>a_{1} = \frac{a_{0}}{2(l+1)}, </math>
+:<math>a_{2} = \frac{a_{1} + \left(\sqrt{k}(l+\frac{3}{2})-E_{l}\right)a_{0}}{2(2l+3)} = \frac{a_{0}}{2(2l+3)}\left(\frac{1}{2(l+1)}+\sqrt{k}\left(l+\frac{3}{2}\right)-E_{l}\right),</math>
+:<math>a_{3} = \frac{a_{2} + \left(\sqrt{k}(l+\frac{5}{2})-E_{l}\right)a_{1}}{6(l+2)},</math>
+:<math>a_{n+1} = \frac{a_{n} + \left(\sqrt{k}(l+\frac{1}{2}+n)-E_{l}\right)a_{n-1}}{(n+1)(2l+2+n)}. </math>
+The two solutions to the indicial equation are <math>m = l+1</math> and <math>m = -l</math> of which the former is taken as it yields the regular (bounded, [[Normalisable wave function|normalizable]]) wave function. For a simple solution to exist, the infinite series is sought to terminate and it is here where particular values of ''k'' are exploited for an exact closed-form solution. Terminating the polynomial at any particular order can be accomplished with different values of ''k'' defining the Hamiltonian. As such there exists an infinite number of systems, differing only in the strength of the harmonic containment, with exact ground-state solutions. Most simply, to impose ''a''<sub>''k''</sub> = 0 for ''k'' ≥ 2, two conditions must be satisfied:
+::<math>\frac{1}{2(l+1)}+\sqrt{k}\left(l+\frac{3}{2}\right)-E_{l} = 0,</math>
+::<math>\sqrt{k}(l+\frac{5}{2})=E_{l}.</math>
+These directly force ''a''<sub>2</sub> = 0 and ''a''<sub>3</sub> = 0 respectively, and as a consequence of the three term recession, all higher coefficients also vanish. Solving for <math>\sqrt{k}\,</math> and <math>E_{l}\,</math> yields
+:<math>\sqrt{k} = \frac{1}{2(l+1)},</math>
+:<math>E_{l} = \frac{2l+5}{4(l+1)},</math>
+and the radial wave function
+:<math>T_{l} = u^{l+1}\left(a_{0}+\frac{a_{0}}{2(l+1)}u\right).</math>
+Transforming back to <math>R_{l}(u)\,</math>
+:<math>R_{l}(u) = \frac{T_{l}(u)e^{-\frac{\sqrt{k}}{4}u^{2}}}{u} = u^{l}\left(1+\frac{1}{2(l+1)}u\right)e^{-\frac{\sqrt{k}}{4}u^{2}},</math>
+the ground-state (with <math>l=0\,</math> and energy <math>5/4 E_{\mathrm{h}}\,</math>) is finally
+:<math>\Phi(\mathbf{u}) = \left(1+\frac{u}{2}\right)e^{-u^{2}/4}.</math>
+Combining, normalizing, and transforming back to the original coordinates yields the ground state wave function:
+:<math>\Psi(\mathbf{r}_{1},\mathbf{r}_{2}) = \frac{1}{2\sqrt{8\pi^{5/2}+5\pi^{3}}}\left(1+\frac{1}{2}|\mathbf{r}_{1}-\mathbf{r}_{2}|\right)\exp\left(-\frac{1}{4}\big(r_{1}^{2}+r_{2}^{2}\big)\right).</math>
+The corresponding ground-state total energy is then <math>E=E_R+E_u=\frac{3}{4}+\frac{5}{4} = 2 E_{\mathrm{h}}</math>.
+==Remarks==
+The exact ground state [[electronic density]] of the Hooke atom is<ref name="kais93"/>
+:<math>\rho(\mathbf{r}) = \frac{2\pi^{3/2}}{(8+5\sqrt{\pi})}e^{-(1/2)r^{2}}\left(\left(\frac{\pi}{2}\right)^{1/2}\left(\frac{7}{4}+\frac{1}{4}r^{2}+\left(r+\frac{1}{r}\right)\mathrm{erf}\left(\frac{r}{\sqrt{2}}\right)\right)+e^{-(1/2)r^{2}}\right).</math>
+From this we see that the radial derivative of the density vanishes at the nucleus. This is in stark contrast to the real (non-relativistic) helium atom where the density displays a cusp at the nucleus as a result of the unbounded Coulomb potential.
+==See also==
+*[[List of quantum-mechanical systems with analytical solutions]]
+==References==
+{{reflist}}
+==Further reading ==
+* {{cite journal | author = Cioslowski, Jerzy | coauthors = Pernal, Katarzyna | title = The Ground State of Harmonium | journal = The [[Journal of Chemical Physics]] | year = 2000 | volume = 113 | pages = 8434–8443 | doi = 10.1063/1.1318767 | issue = 19|bibcode = 2000JChPh.113.8434C }}
+* {{cite journal | author = O’Neill, Darragh P. | coauthors = Gill, Peter M. W. | title = Wave functions and two-electron probability distributions of the Hooke's-law atom and helium |url=http://rsc.anu.edu.au/~pgill/papers/084Hookium.pdf | year = 2003 | journal = [[Physical Review A]] | volume = 68 | issue = 2 | page = 022505 | doi = 10.1103/PhysRevA.68.022505|bibcode = 2003PhRvA..68b2505O }}
+[[Category:Quantum chemistry]]
+[[Category:Quantum mechanics]]
+[[Category:Quantum models]]

Fuchs's theorem: Difference between revisions

Latest revision as of 08:43, 19 March 2013

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Fuchs's theorem: Difference between revisions

Latest revision as of 08:43, 19 March 2013

Definition

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Remarks

See also

References

Further reading

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