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To be self-consistent |
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| Mar11-05, 02:31 AM | #1 |
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To be self-consistent
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I am wondering what it means for a solution (or theory) to be\nself-consistent. My question comes from my current studies of the\nSelf-Consistent Hartree-Fock Method. The so called Hartree Equation\ncan be solved in a "self-consistent manner".\n\nEither general comments about self-consistent theories or comments in\nthe conext of Hartree-Fock would be very useful.\n\nThanks,\nBill\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I am wondering what it means for a solution (or theory) to be
self-consistent. My question comes from my current studies of the Self-Consistent Hartree-Fock Method. The so called Hartree Equation can be solved in a "self-consistent manner". Either general comments about self-consistent theories or comments in the conext of Hartree-Fock would be very useful. Thanks, Bill |
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| Mar12-05, 08:08 AM | #2 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>billkavanagh@gmail.com wrote:\n> I am wondering what it means for a solution (or theory) to be\n> self-consistent. My question comes from my current studies of the\n> Self-Consistent Hartree-Fock Method. The so called Hartree Equation\n> can be solved in a "self-consistent manner".\n>\n> Either general comments about self-consistent theories or comments in\n> the conext of Hartree-Fock would be very useful.\n\nIt just means that you have two sets of equations relating two\nsets of unknown quantities, and you want to solve the equations\njointly for the unknowns.\n\nIf aspect A of a theory says y=x^2 and aspect B of the theory\n(or of another theory) says x=y-2 then self-consistency means that\nboth equations are assumed to be valid, giving\nx^2 = y = x+2,\nwhich leads to the two solutions\nx=2, y=4 and x=-1, y=1.\n\nThat\'s all. Of course, the Self-Consistent Hartree-Fock Method\nis harder to solve, but the principle is the same.\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>billkavanagh@gmail.com wrote:
> I am wondering what it means for a solution (or theory) to be > self-consistent. My question comes from my current studies of the > Self-Consistent Hartree-Fock Method. The so called Hartree Equation > can be solved in a "self-consistent manner". > > Either general comments about self-consistent theories or comments in > the conext of Hartree-Fock would be very useful. It just means that you have two sets of equations relating two sets of unknown quantities, and you want to solve the equations jointly for the unknowns. If aspect A of a theory says [itex]y=x^2[/itex] and aspect B of the theory (or of another theory) says [itex]x=y-2[/itex] then self-consistency means that both equations are assumed to be valid, giving [itex]x^2 = y = x+2,[/itex] which leads to the two solutions [itex]x=2, y=4[/itex] and [itex]x=-1, y=1[/itex]. That's all. Of course, the Self-Consistent Hartree-Fock Method is harder to solve, but the principle is the same. Arnold Neumaier |
| Mar14-05, 01:11 AM | #3 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE><billkavanagh@gmail.com> a écrit dans le message de\nnews:1110422232.529994.198820@l41g2000cwc.googlegroups.com...\n\n&g t; I am wondering what it means for a solution (or theory) to be\n> self-consistent. My question comes from my current studies of the\n> Self-Consistent Hartree-Fock Method. The so called Hartree Equation\n> can be solved in a "self-consistent manner".\n>\n> Either general comments about self-consistent theories or comments in\n> the conext of Hartree-Fock would be very useful.\n\nThere are two meanings. The most common one is simply logical\nconsistency, and applies only to theories. That is, from a set of\naxioms or first principles of the theory, one can\'t prove the contrary\nto another axiom or first principle of the same theory.\n\nThe meaning here is still simpler. It is that the field in which the\nstates of the particles is calculated is the same as the one calculated\nin turn from this states. Differently said, the solution doesn\'t\ncontradict the hypothesis.\n\n--\n~~~~ clmasse on free F-country\nLiberty, Equality, Profitability.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky><billkavanagh@gmail.com> a écrit dans le message de
news:1110422232.529994.198820@l41g20...egroups.com... > I am wondering what it means for a solution (or theory) to be > self-consistent. My question comes from my current studies of the > Self-Consistent Hartree-Fock Method. The so called Hartree Equation > can be solved in a "self-consistent manner". > > Either general comments about self-consistent theories or comments in > the conext of Hartree-Fock would be very useful. There are two meanings. The most common one is simply logical consistency, and applies only to theories. That is, from a set of axioms or first principles of the theory, one can't prove the contrary to another axiom or first principle of the same theory. The meaning here is still simpler. It is that the field in which the states of the particles is calculated is the same as the one calculated in turn from this states. Differently said, the solution doesn't contradict the hypothesis. -- ~~~~ clmasse on free F-country Liberty, Equality, Profitability. |
| Mar15-05, 12:21 PM | #4 |
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To be self-consistent
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Fri, 11 Mar 2005 08:31:56 +0000, billkavanagh wrote:\n\n> I am wondering what it means for a solution (or theory) to be\n> self-consistent. My question comes from my current studies of the\n> Self-Consistent Hartree-Fock Method. The so called Hartree Equation can\n> be solved in a "self-consistent manner".\n>\n> Either general comments about self-consistent theories or comments in the\n> conext of Hartree-Fock would be very useful.\n\nSelf consistency is simple to explain. Suppose that you want to calculate\nquantity A, but to make any progress you must make simplifying assumptions\nabout A. Given these assumptions, you may be able to derive an equation of\nthe form A = f(A). Solving this equation gives you a self consistent\napproximation to A.\n\nThe prototypical example is a spin system. You assume that the magnetic\nfield produced by the neighboring spins at each site is spatially uniform.\nThis allows you to calculate the energy of each spin site. Which allows\nyou to calculate the partition function. Which allows you to calculate the\naverage magnetic field at each site. You\'ve arrived at an expression for\nthe local magnetic field in terms of itself. If you solve this equation\nyou get an answer that is consistent with your assumption of spacial\nuniformity.\n\nIn the case of Hartree-Fock, the initial assumption that you make is that\nthe wave function of of each electron looks the same (note that\nantisymmetrization here is important). You use this to calculate the\npotential that each electron feels due to all the other ones. Plug this\npotential into the Schroedinger equation and solve. There are various\nother ways of formulating the Hartree-Fock approximation, but the essence\nis here.\n\nHope this helps.\n\nIgor\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Fri, 11 Mar 2005 08:31:56 [itex]+0000,[/itex] billkavanagh wrote:
> I am wondering what it means for a solution (or theory) to be > self-consistent. My question comes from my current studies of the > Self-Consistent Hartree-Fock Method. The so called Hartree Equation can > be solved in a "self-consistent manner". > > Either general comments about self-consistent theories or comments in the > conext of Hartree-Fock would be very useful. Self consistency is simple to explain. Suppose that you want to calculate quantity A, but to make any progress you must make simplifying assumptions about A. Given these assumptions, you may be able to derive an equation of the form [itex]A = f(A)[/itex]. Solving this equation gives you a self consistent approximation to A. The prototypical example is a spin system. You assume that the magnetic field produced by the neighboring spins at each site is spatially uniform. This allows you to calculate the energy of each spin site. Which allows you to calculate the partition function. Which allows you to calculate the average magnetic field at each site. You've arrived at an expression for the local magnetic field in terms of itself. If you solve this equation you get an answer that is consistent with your assumption of spacial uniformity. In the case of Hartree-Fock, the initial assumption that you make is that the wave function of of each electron looks the same (note that antisymmetrization here is important). You use this to calculate the potential that each electron feels due to all the other ones. Plug this potential into the Schroedinger equation and solve. There are various other ways of formulating the Hartree-Fock approximation, but the essence is here. Hope this helps. Igor |
| Apr9-05, 02:57 AM | #5 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>The Contraction mapping theorem will be useful to understand this\nterm. It is the underlying principle of all iterative procedures.\nIn your case the matrix elements of the Hamiltonian (for the\nHartree-Fock method it is the Fock matrix) depend on the density.\nThus, you must "force" the\nsystem to be "self-consistent". Or following the previous suggestion\nyou may rewrite the one-electron Density in terms of the Fock\noperator, which itself depends on the density. It can be done by\ninverting the fock operator and getting the density from the set of\none-electron orbitals, eigenvectors. You follow the convergence\nparameter during the solution process (Roothan-Hall procedure, which\nstarts from the zero guess - or core hamiltonian) which is the density\ngradient. Eventually you force dE/dC to be zero. Hope this helps.\n\nAqyl.\nbillkavanagh@gmail.com wrote in message news:<1110422232.529994.198820@l41g2000cwc.googlegroups.com>...\n > I am wondering what it means for a solution (or theory) to be\n> self-consistent. My question comes from my current studies of the\n> Self-Consistent Hartree-Fock Method. The so called Hartree Equation\n> can be solved in a "self-consistent manner".\n>\n> Either general comments about self-consistent theories or comments in\n> the conext of Hartree-Fock would be very useful.\n>\n> Thanks,\n> Bill\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>The Contraction mapping theorem will be useful to understand this
term. It is the underlying principle of all iterative procedures. In your case the matrix elements of the Hamiltonian (for the Hartree-Fock method it is the Fock matrix) depend on the density. Thus, you must "force" the system to be "self-consistent". Or following the previous suggestion you may rewrite the one-electron Density in terms of the Fock operator, which itself depends on the density. It can be done by inverting the fock operator and getting the density from the set of one-electron orbitals, eigenvectors. You follow the convergence parameter during the solution process (Roothan-Hall procedure, which starts from the zero guess - or core hamiltonian) which is the density gradient. Eventually you force [itex]dE/dC[/itex] to be zero. Hope this helps. Aqyl. billkavanagh@gmail.com wrote in message news:<1110422232.529994.198820@l41g2...groups.com>... > I am wondering what it means for a solution (or theory) to be > self-consistent. My question comes from my current studies of the > Self-Consistent Hartree-Fock Method. The so called Hartree Equation > can be solved in a "self-consistent manner". > > Either general comments about self-consistent theories or comments in > the conext of Hartree-Fock would be very useful. > > Thanks, > Bill |
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