We know that there is convergence issue of DFT method on study real molecular system. It would be conducive if people could pre-determine if the SCF procedure is converge analytically before starting the calculation.
This question is a bit ill-defined: what do you mean by "the self-consistent field procedure"? If you mean the original Roothaan procedure, then the question makes sense, but it is uninteresting: nobody uses the Roothaan procedure, since it usually doesn't converge, and you need to do something smarter like use damping or other convergence acceleration schemes.
But, these are different methods, and now you would have to study each of them separately.
Still, it is possible to make any self-consistent field calculation converge simply by switching from iterative diagonalization to direct energy minimization. Here, you rewrite the problem in terms of iterative orbital rotations, and what you get is the minimization of a scalar function f(theta) in Cartesian space, which is a well-understood problem in numerical analysis. There are methods for minimization without gradients (e.g. the Nelder-Mead "amoeba" method), with gradients (e.g. steepest descent and conjugate gradients, and preconditioned versions thereof), and with Hessians (e.g. Newton-Raphson and trust region methods). These methods are proven to always converge to an extremum, and you'll just need to check whether you're at a local minimum or not just as if you use some kind of iterative diagonalization.
For details, you can refer to our recent open access overview paper: Molecules 2020, 25 (5), 1218.
If by "the SCF method" you mean the simple SCF, the answer is: no, usually it does not converge (unless the problem is very simple; basically the gap is huge, so that the system does not respond very much to potentials). The damped SCF problem on the other hand does converge, for damping parameters small enough (unless you run into fractional occupations). The trouble is that it's not trivial to find out a priori what is a good damping parameter, so you have to couple that with a globalization strategy (like line searches). A good review paper is https://arxiv.org/abs/1905.02332, and for technical aspects about SCF/DM convergence I will shamelessly cite a recent preprint of mine: https://arxiv.org/abs/2004.09088